Numerical Solution and Exponential Decay to Von Kármán System with Frictional Damping

Transcription

1 Appl. Mah. Inf. Sci. 8, No. 4, (4) 575 Applied Mahemaics & Informaion Sciences An Inernaional Jornal hp://d.doi.org/.785/amis/84 Nmerical Solion and Eponenial Deca o Von Kármán Ssem wih Fricional Damping D. C. Pereira, C. A. Raposo and J. A. J. Avila, Deparmen of Mahemaics, Universidade do Esado do Pará, , Belém, PA, Brazil Deparmen of Mahemaics and Saisics, Universidade Federal de São João del Rei, 67-94, São João del Rei, MG, Brazil Received: Jl., Revised: 4 Oc., Acceped: 5 Oc. Pblished online: Jl. 4 Absrac: In his work we consider he Von Kármán ssem wih fricional damping acing on he displacemen and sing he Mehod of Nakao we prove he eponenial deca of he solion. The nmerical scheme is presened for calclae he solion and o verif he long-ime deca energ. Kewords: Von Kármán ssem, Mehod of Nakao, Deca of solions, Nmerical solion, Finie differences mehod. Inrodcion For several ears he ssem of Theodor von Kármán [9] was sdied in differen siaions and mehods. The eponenial deca of he energ o he von Kármán eqaions wih memor in nonclindrical domains was sdied b Park and Kang [7] in 9 sing he same mehod as in [8]. To he models of von Kármán aking ino accon for roaional forces, Bradle and Lasiecka [6] in 994 showed he niform deca raes for he solions in clindrical domain The niform deca of he solion was considered for fricional dissipaive a he bondar, for eample, in he works of Horn and Lasiecka [8] in 994, Horn and Lasiecka [9] in 995, and Horn, Favini, Lasiecka and Taar [7] in 996. Appling mlipliers mehod Avalos and Lasiecka [] in 987 showed he niform deca for wo-dimensional linear hermoelasic plaes and Avalos and Lasiecka [] in 998 he one-dimensional hermoelasic von Kármán model was sdied. For hermal damping Menzala and Zaza [] in 998 proved he eponenial deca b he semigrop properies. For Viscoelasic plaes wih memor, sing energ mehod, we cie Rivera and Menzala [5] in 999, and he work of Rivera, Oqendo and Sanos [6] in 5 where was proved ha he energ decas niforml, eponeniall or algebraicall wih he same rae of deca of he relaaion fncion. Based on mlipliers mehod, he eponenial deca of solion for he fll von Kármán Ssem of Dnamic Thermoelasici was proved b Benabdallah and Lasiecka [4] in. For von Kármán Ssem wih memor Raposo and Sanos [] in obained he General Deca of solion sing he idea of Messaodi [] in he sd of he asmpoic behavior of viscoelasic eqaions. For he nmerical scheme we menion for eample Reinhar [4] in 98 where was sdied he approimaion of he von Kármán eqaions saionar b he mied finie elemen. The work of Yosibash, Kirb and Golieb [] in 4 where was sdied he von Kármán ssem over recanglar domains and nmericall solved sing boh he Chebshev-collocaion and Legendre-collocaion mehods for he spacial discreizaion and he implici Newmark-β scheme combined wih a non-linear fied poin algorihm for he emporal discreizaion, and Bilbao [5] in 7 sed nmerical sabili for nmerical mehods for he von Kármán ssem, hrogh he se of energ-conserving mehods. Wha disingishes his paper from oher relaed works is ha we appl he Mehod of Nakao in he von Kármán ssem o prove he eponenial deca of he solion and we presen an nmerical scheme b finie Corresponding ahor avila jaj@fsj.ed.br c 4 NSP Naral Sciences Pblishing Cor.

2 576 D. C. Pereira e. al. : Nmerical Solion and Eponenial Deca o... differences mehod o nmerical solion and he long-ime deca energ, in his sense, here is few resl in he lierare. The remainder of his paper is organized as follows. In secion we presen he resl of eisence of weak solion, in he secion we prove he eponenial deca of he solion, in he secion 4 we applied he Finie Difference Mehod in he von Kármán ssem and finall in he secion 5 we give he conclsion. Eisence of solion We se he sandard Lebesge space and Sobolev Space wih heir sal properies as in Adams (975) [] and in his sense (, ) and, denoe he inner prodc in L and H respecivel. B we denoe he sal norm in L. Le Ω R be a bonded domain of he plane wih reglar bondar Γ. For a real nmber T > we denoe Q=Ω (,T) and Σ = Γ (,T). Here =(,,) is he displacemen, v=v(,,) he Air sress fncion and η is he ni normal eernal in Ω and =. Wih his noaion we have he following ssem and where [,v]+ = f in Q, () v+[,]= in Q, () ()=, ()= in Ω, () =v=, η = v = on Σ, (4) η [,v]= v v + v Now sing he same idea as in [] we have he following resl of eisence of solion. Theorem. For H (Ω), L (Ω) and f L loc (R+ ;L (Ω)) here eiss,v : Q R sch ha,v L (,T ;H (Ω)), L (,T ;L (Ω)), and, v weak solion ()-(4). Asmpoic behavior In his secion, we will se he Mehod of Nakao (978) [] o prove he eponenial deca of he solion. Firs we define E()= () + () + v() (5) Lemma. The fncional of energ E() is limied. Proof. Mlipling () b and inegraing in Ω, we have d [ () + () ] [(),v()], () d + () =( f(), ()) Using () we obain [(),v()], () = [(), ()], v() = d [(),()], v() d v (),v() = d 4 d v() from where follows d [ () + () + ] d v() + () = ( f(), ()) (6) Performing inegraion from o follows b Cach-Schwarz ineqali we obain hen () + () + v() + (s) ds (s) ds+ f(s) ds+ + + v E()+ (s) ds E()+ f(s) ds from where follows E() C wih C consan independenl of. Now we inrodce a new fncional. Lemma. The fncional saisfies + F ()=E() E(+ )+ f(s) ds + (s) ds F () Proof. Inegraing (6) from τ o τ wih <τ < τ, we obain E(τ )+ τ and for all > + E(+ )+ τ (s) ds = E(τ ) + τ + (s) ds=e()+ + E()+ f(s) ds+ τ ( f(s), (s))ds (7) ( f(s), (s))ds + (s) ds c 4 NSP Naral Sciences Pblishing Cor.

3 Appl. Mah. Inf. Sci. 8, No. 4, (4) / hen + (s) ds E() E(+ ) + Lemma. The fncional + G ()=8C esssp (s) F()+(+C ) s [,+] saisfies Proof. Firs we noe ha f(s) ds = F () (8) () d +C f() d ( () + v() ) d G () [(),v()], () = [(),()], v() = v(), v() = v() (9) From (8) here eiss [,+ 4 ] and [+ 4,+ ] sch ha ( i ) F(), i=, () Mlipling () b and inegraing in Ω, we have d d ( (),()) () + () [(),v()], () +( (),())=( f(),()) Performing inegraion from o and sing (9) we have ( () + v() ) d =( ( ),( )) ( ( ),( ))+ ( () ( (),()) ) d + ( f(),())d Now, choosing C sch ha C and appling Cach-Schwarz ineqali we ge ( ) () + v() d ( C esssp (s) ( ( ) + ( ) ) ) sing (), +(+C ) s [,+] () d+c f() d, ( ) () + v() d 8C esssp (s) F()+(+C ) s [,+] () d +C f() d, and hen ( () + v() ) d G () () Theorem. For f Lloc (R+ ;L (Ω)) wih f(s) ds α e α, for all and α,α >, hen he solion (, v) saisfies () + () + v() + + (s) ds k e k, () for almos ever, wih k,k >, consans independenl from. Proof. From (8) and () we concldes ( () + () + ) v() d F ()+G () There is [, ] sch ha E( ) = ( ) + ( ) + v( ) From (7) we ge (F ()+G ()) () E( )=E( )+ (s) ds ( f(s), (s))ds Then E() = E( )+ and E( )+ + (s) ds (s) ds+ + ( f(s), (s))ds f(s) ds, + + ess sp E(s) E( )+ (s) ds+ f(s) ds s [,+] Now sing (8) and () we obain ess sp E(s) (F ()+G ())+F ()+ s [,+] + 5F ()+6Cesssp (s) F() s [,+] + + 4(+C ) (s) ds + (+4C ) + f(s) ds (9+4C )F ()+ esssp E(s) + 8C F ()+(+4C ) s [,+] + f(s) ds f(s) ds, c 4 NSP Naral Sciences Pblishing Cor.

4 578 D. C. Pereira e. al. : Nmerical Solion and Eponenial Deca o... from where follows ess sp E(s) (74+8C )F ()+(+8C ) s [,+] and hen + + E() C (E() E(+ ))+C f(s) ds, where C i, i=,, consans independenl from. f(s) ds, Wiho los of generali, we can sppose C > and for <β = C < we, have + E(+ ) ( β)e()+βc f(s) ds For and n N sch ha n n+ Now E() ( β)e( )+βc ( β) n E( n)+βc n f(s) ds f(s) ds ( β) n+ <( β) implies ( β) n <( β) Then E() ( β) esssp s [,] = m β ( β) + βc wih m = essspe(s)<. s [,] Now we have E(s)+βC f(s) ds E() < m β e ln( β) + βc f(s) ds, < m β e β + βc α e α, f(s) ds for almos ever, wih β = ln( β)> and hen 4 Nmerical solion For a given small consan ε > we define a hin plae b Ω ( ε,ε)={(,,z) R :(,) Ω, z ( ε,ε)} whose midsrface is idenified wih Ω. We resolve he von Kármán ssem in a sqare hin elasic plae b Finie Difference Mehod, sbjeced o a perpendiclar load f and bondar condiion of clamped pe. 4. Discree formlaion Consider he discree domain he midsrface of he sqare plae, Ω h = (,π) wih niform grid i = ih, j = jh, i, j =,...,N +, h = π/(n + ). The inernal poin are i = ih, j = jh, i, j N. The bondar of Ω h is denoed Γ h. The emporal discreizaion of inerval I k = (,T) is given b n = nk, n=,...,m+, k= C h/ where C is a posiive consan and T = k(m+ ). Denoe b n i, j and v n i, j he fncions and v evalae in he poin ( i, j ) and a he insan n, respecivel. I also, denoed b Q k h = Ω h I k and Σ k h = Γ h I k. We show in Figre he paern mesh of Ω wih is poins: inernal (circles), bondar (sqares) and ghos (diamonds). We define he following discree differenial () + () + v() < γ e γ (4) From (8), we have + + (s) ds E()+ f(s) < γ e γ 4 (5) wih γ i > consans. Finall we concldes from (4) and (5) ha here is consans k,k > sch ha () + () + + v() + (s) ds<k e k This complees he proof. Fig. : The paern mesh of Ω wih inernal, bondar and ghos poins. operaors: δ n i, j = k (n i, j n i, j ), δ n i, j = k (n+ i, j n i, j+ i, n j ) (6) c 4 NSP Naral Sciences Pblishing Cor.

5 Appl. Mah. Inf. Sci. 8, No. 4, (4) / δ n i, j = δ (δ n i, j)= δ n i, j = h (n i+, j n i, j+ n i, j ), δ n i, j = h (n i, j+ n i, j+ n i, j ) (7) 4h (n i+, j+ n i+, j n i, j+ + n i, j ) (8) δ 4 n i, j = δ (δ n i, j)= h 4(n i+, j 4 n i+, j+ 6 n i, j 4 n i, j+ n i, j) (9) δ 4 n i, j = δ (δ n i, j)= h 4(n i, j+ 4 n i, j++ 6 n i, j 4 n i, j + n i, j ) () δ n i, j = δ (δ n i, j)= h 4(n i+, j+ n i+, j+ n i+, j n i, j++ 4 n i, j n i, j + n i, j+ n i, j + n i, j ) () The discree biharmonic operaor is given b h n i, j = δ 4 n i, j+ δ 4 n i, j+ δ n i, j () The discree bracke operaor is given b [ n i, j, v n i, j] h = δ n i, j δ v n i, j+ δ n i, j δ v n i, j δ n i, j δ v n i, j () Using he eqaions (6) o () we obain he discree model of von Kármán ssem, δ n i, j a h n i, j a [ n i, j, v n i, j] h + a δ n i, j = f n i, j in Q k h, (4) h vn i, j+[ n i, j, n i, j] h = g n i, j in Q k h, (5) i, j =( ) i, j, δ i, j =( ) i, j in Ω h (6) ( ) n = η i, j n i, j = v n i, j = on Σh k ( ), (7) v n = on Σh k η, (8) where a,a and a are consans. a is he damping parameer, f and g are perpendiclar and horizonal loads, respecivel. Noe ha we obain he discree model of eqaions () o (4) when a = a = a = and g=. For he emporal compaion of he eqaions (4) o (8), we have splied in hree levels:, and. For calclae in he level, we se (7), i.e., i, j i, j =( ) i, j (9) For calclae in he level, we se (7) and (9), i.e., i, j = i, j+ k( ) i, j () For calclae in he level, we firs calclae h vn i, j = [ n i, j, n i, j] h + g n i, j, n=,...,m. () Using () for fncion v, he eqaion () is given b Av n i, j = Bv n i, j + Dv n i, j [ n i, j, n i, j] h + g n i, j, n=,...,m () where A = (a i, j ) N N is a smmeric mari, B = (b i, j ) N N and D = (d b i, j ) N N g, where N b is he nmber he bondar poins, N g is he nmber he ghos poins and, v and v denoe he fncion v evalae in he bondar and ghos poins, respecivel. Ths, for all n=,...,m v n = () v n = v n (4) The eqaions () and (4) are de o a clamped bondar condiion, given b (7) and (8), and becase he eerior normal coincides wih canonical vecors. Ths, he linear ssem () is resolved b he SOR mehod. Once known v we can calclae for all n=,...,m where, n+ i, j = µ ũ n i, j+ µ ( ω n i, j (/8)ω n i, j + ω n i, j + µ n i, j+ µ 4 n i, j + µ 5 f n i, j (5) µ = a k /h 4, µ = a k /h 4, µ = a k µ 4 = a k, µ 5 = k, ũ n i, j = n i+, j + n i+, j+ 8n i+, j + n i+, j + n i, j+ 8 n i, j+ + n i, j 8 n i, j + n i, j + n i, j+ 8 n i, j + n i, j + n i, j, ω n i, j = (n i+, j n i, j+ n i, j )(vn i, j+ vn i, j+ v n i, j ), ω n i, j = (n i+, j+ n i+, j n i, j+ + n i, j ) (v n i+, j+ vn i+, j vn i, j+ + vn i, j ), ω n i, j = ( n i, j+ n i, j+ n i, j )(v n i+, j v n i, j+ v n i, j) 4. Nmerical ess Consider he following analical solion of he eqaions (4) - (8) wih loads given b a (,,) = sin sin e (6) v a (,,) = sin sin (7) f(,,) = e [ ( a )sin sin + 8a (cossin coscos+cossin ) ] a (4cossin cossin sin sin ) g(,,) = 8(cossin coscos+cossin ) ) (8) +e (4cossin cossin sin sin ) (9) c 4 NSP Naral Sciences Pblishing Cor.

6 58 D. C. Pereira e. al. : Nmerical Solion and Eponenial Deca o... We se for size of mesh N = 9. For case g = he mechanical energ is given b E()= () + a () + a v() (4) Eample. The firs problem we consider he following daa: a =.,a =.5 and a =.5. f and g are given in (8) and (9), respecivel. and are obained from (6) and and v saisfies he clamping condiions. For consan C =.4 a convergence is aained in T = k(m+ ).46(8) 6.75 s. In Figre we presen for all (,T], he absole error defined b () a (). In Figre, we show he long-ime behavior of he ransversal displacemen in he poin (π/,π/). In Figre 4, we show in D he ransversal displacemen of plae for differen ime seps: =, 5.785, 7.99, We iniiall observe ha he deflecions are larger in he corners of he plae and hen he redce smoohl and epand rapidl near he bondar (a) (b) Absole Error Fig. : The absole error a L norm he long-ime (c) (d) Fig. : Transversal displacemen in he poin (π/,π/). Fig. 4: Transversal displacemen of plae. (a) = s, (b) s, (c) 7.99 s and (d) s. Eample. In his eample we consider a =.,a = and a =. f is given in (8) and g=. c 4 NSP Naral Sciences Pblishing Cor.

7 Appl. Mah. Inf. Sci. 8, No. 4, (4) / 58 and are obained from (6) and and v saisf he clamping condiions. For consan C =. a convergence is aained in T = k(m + ).459(86) s. For calclae of energ of ssem, given b (4), we have compaed in all he plae sing he Composie Simpson s rle. In Figre 5 and 6 we show he long-ime behavior of he solion in he poin (π/, π/) and energ of ssem, respecivel. Noe ha energ converge o. In Figre 7, we show he ransversal displacemen of he plae, in D, for differen ime seps: =, 45.44, , We iniiall observe ha he deflecions are larger near he bondar, b in long-ime hese deflecions disappear (b) (c) Fig. 5: Transversal displacemen in he poin (π/,π/)..5 Energ (d) Fig. 7: Transversal displacemen of plae. (a) = s, (b) s, (c) and (d) s Fig. 6: The energ of ssem hrogh long-ime. 5 Conclsion The Nakao s mehod proved o be an efficien mehod for he demonsraion of he eponenial deca of he solion of he ssem of von Kármán. Nmerical ess have shown he deca of he mechanical energ of he ssem..5 References (a) [] Adams, R. A.; Academic Press, New York, (975). [] Avalos, G.; Lasiecka, I.; Eponenial sabili of a hermoelasic plaes wih free bondar condiions and wiho mechanical dissipaion, SIAM J. Mah. Anal., 9, 55-8 (998). c 4 NSP Naral Sciences Pblishing Cor.

8 58 D. C. Pereira e. al. : Nmerical Solion and Eponenial Deca o... [] Avalos, G.; Lasiecka, I.; Eponenial sabili of a hermoelasic ssem wiho mechani- cal dissipaion, Rend. Isi. Ma. Univ. Triese., 8, -8 (987). [4] Benabdallah, A.; Lasiecka, I.; Eponenial Deca Raes for a Fll von Kármán Ssem of Dnamic Thermoelasici, Jornal of Differenial Eqaions, 6, 5-9 (). [5] Bilbao, S.; A Famil of Conservaive Finie Difference Schemes for he Dnamical von Kármán Plae Eqaions, Nmerical Mehods for Parial Differenial Eqaions, 4, 9-6 (7). [6] Bradle, M. E.; and Lasiecka, I.; Global deca raes for he solions o a von Kármán plae wiho geomeric condiions, J. Mah. Anal. Appl., 8-54 (994). [7] Horn, M.; Favini, A.; Lasiecka, I.; Taar, D.; Global eisence, niqeness and reglari o a Von Kármán ssem wih nonlinear bondar dissipaion, Diff. and In. Eq., 9, (996). [8] Horn, M.; Lasiecka, I.; Uniform deca of weak solions o a von Kármán plae wih nonlinear bondar dissipaion, Diff. and In. Eqaions, 7, (994). [9] Horn, M.; Lasiecka, I.; Global sabilizaion of a dnamical Von Kármán plae wih Nonlinear Bondar Feedback, Appl. Mah. Opimizaion,, (995). [] Menzala, G. P.; Zaza, E.; Energ deca raes for he von Kármán ssem of hermoelasic plaes, Differenial and Inegral Eqaions,, (998). [] Messaodi, S.A.; General deca of solions of a viscoelasic eqaion, Jornal of Mahemaical Analsis and Applicaions, 4, (8). [] Nakao, M.; A difference ineqali and is applicaion o nonolinear evolion eqaion, J. Mah. Soc. Japan,, (978). [] Raposo, C. A.; Sanos, M. L.; General deca o a von Kármán Ssem wih memor, Nonlinear Analsis, 74, (). [4] Reinhar, L.; On he Nmerical Analsis of he Von Kármán Eqaions: Mied Finie Elemen Approimaion and Coninaion Techniqes, Nmer. Mah., 9, 7-44 (98). [5] Rivera, J. E. M.; Menzala, G.P.; Deca raes of solions of a von Kármán ssem for viscoelasic plaes wih memor, Qarerl of Applied Mahemaics. v. LVII,, 8- (999). [6] Rivera, J. E. M.; Oqendo, H. P.; Sanos, M. L.; Asmpoic behavior o a von Kármán plae wih bondar memor condiions, Nonlinear Analsis, 6, 8-5 (5). [7] Park, J. Y.; Kang, J. R.; Global eisence and sabili for a von Kármán eqaions wih memor in nonclindrical domains, J. Mah. Phs., 5, -7 (9). [8] Sanos, M. L.; Ferreira, J.; Raposo, C. A.; Eisence and niform deca for a nonlinear beam eqaion wih nonlineari of Kirchhoff pe in domains wih moving bondar, Absr. Appl. Anal., 9-99 (5). [9] von Kármán, T.; Fesigkeisprobleme im Maschinenbam, Encklopadie der Mah. Wiss. V/4C, Leipzig, -85 (9). [] Yosibash, Z.; Kirb, R.M.; Golieb, D.; Collocaion mehods for he solion of von-kármán dnamic non-linear plae ssems, Jornal of Compaional Phsics,, 4-46 (4). Dcival C. Pereira, PhD in Sciences from Federal Universi of Rio de Janeiro and pos-docoral in Mahemaics b he Naional Laboraor of Scienific Comping (LNCC). Professor of Pará Sae Universi and researcher in Mahemaics, wih emphasis on analsis, working mainl in he solion problems of nonlinear hperbolic parial differenial eqaions. Carlos Albero Raposo da Cnha, PhD in Mahemaics from UFRJ - Federal Universi of Rio de Janeiro. Fll Professor a he Deparmen of Mahemaics and Saisics from UFSJ - Federal Universi of São João del-rei. Has eperience in Mahemaics wih emphasis in Parial Differenial Eqaions. The ineress are mainl in deca of energ for dissipaive ssems, ransmission problems, problems in viscoelasici wih memor, hermoelasici and asmpoic behavior of solions of PDEs. Reviewer for he Mahemaical Reviews / AMS - American Mahemaical Socie. Has served as referee for several inernaional mahemaical jornals. Jorge Andrés Jlca Avila, PhD in Mechanical Engineering from USP - Universi of São Palo. Assisan Professor a he Deparmen of Mahemaics and Saisics from UFSJ - Federal Universi of São João del-rei. Has eperience in Applied Mahemaics emphasizing Nmerical Analsis, Fncional Analsis and PDEs. For he momen, he major ineress are in he Navier-Sokes Eqaions, Non Newonian Flids and Finie Elemens. Is also eperienced in Mechanical Engineering wih emphasis in Flid Mechanics and Flid Dnamics. Is member of he Scienific Commiee of UFSJ. c 4 NSP Naral Sciences Pblishing Cor.