Decay of solutions of wave equations with memory

Transcription

1 Proceedings of te 14t International Conference on Computational and Matematical Metods in Science and Engineering, CMMSE July, 14. Decay of solutions of wave equations wit memory J. A. Ferreira 1, P. de Oliveira 1 and G. Pena 1 1 CMUC, Department of Matematics, University of Coimbra s: ferreira@mat.uc.pt, poliveir@mat.uc.pt, gpena@mat.uc.pt Abstract In tis paper we consider a linear damping wave equation wit a memory effect using exponential kernels. We establis a bound for an energy function tat is sown to converge to zero. Numerical waves tat mimic teir continuous counterpart are also introduced using te finite element approac and te qualitative beaviour of te solutions is explored. Key words: wave equation, memory, viscoelasticity, energy estimates 1 Introduction In tis paper we consider te following wave equation wit memory d u t + cdu dt dt t D 1 ut = D K er t s usds + ft, t R +, 1 were K er s = 1 e s, >, ut denotes a function defined from Ω R n into R, c is a function depending only on spatial variables and accounts for te damping of te wave, D 1, D and are positive constants and f denotes a source term. Equation 1 can be used to model te displacement of a viscoelastic material under te action of an external force wen te stress tensor σt and te strain tensor ɛt are related by te following constitutive equation σt = EDɛt Et sdɛs ds, s were D is an elastic tensor and te stress relaxation function, E, is nonnegative and monotone decreasing. Assuming tat te viscoelastic beaviour is described by Maxwell- Wiecert model wit only one Maxwell arm, ten Et = E + E 1 e α 1t, were E is te c CMMSE ISBN:

2 Decay of solutions of wave equations wit memory Young modulus of te spring arm, E 1 is te Young modulus of te Maxwell arm and α 1 = E 1 µ 1 being µ 1 te associated viscosity. Te relation between te displacement u, te stress σ and te external force f is given by Newton s second law ρ d u = σ + f, 3 dt n n were ρ is te mass density of te body and σ = σ ij. Assuming tat te x j strain and te displacement is given by ɛt = 1 ut + ut t, from 3 we obtain for te displacement te following second order integro-differential equation ρ d u dt t D 1 ut = D j=1 i,j=1 K er t s us ds ds + f, 4 wit D 1 = DE + E 1, D = E 1 and = α1 1. In wat follows we consider omogeneous Diriclet boundary conditions and te following initial conditions u = u, du dt = u 1. 5 A quasilinear problem of te type of 1 was also introduced for instance in [5], [9] and [11] to describe a viscoleasticity pysical problem. Witout being exaustive we mention [1], [], [], [7], [8],[1] and [1] for te study of qualitative properties of partial differential problems defined by equations of te type of 1. Te initial boundary value problem IBVP 1, 5 wit omogeneous Diriclet boundary conditions is now replaced by its weak formulation. To define suc formulation we introduce te functional context needed. Let L Ω, L Ω and H 1 Ω be te usual Sobolev spaces. In L Ω we consider te usual inner product, and te norm induced by tis inner product is denoted by. In H 1Ω we consider te usual norm 1. Let L R + ; H 1 Ω be te space of functions v : R+ T H 1Ω suc tat vt 1 dt <, T >. Let H 1 R + ; H 1Ω be te subspace of L R + ; H 1 Ω of all functions v suc tat its weak derivative dv dt : R+ H 1 Ω belongs to L, ; H 1Ω. By L R + ; L Ω we represent te space of all functions v : R + L Ω suc tat ess sup vt <, T >. t [,T ] Let V = L Ω or V = H 1Ω. By Cm R + ; V, m N, we represent te space of function v : R + V wit continuous derivatives dj v dt j : R+ V, for j =,..., m. c CMMSE ISBN:

3 J. A. Ferreira, P. de Oliveira, G. Pena Let u H 1 R + ; H 1 Ω be suc tat d u dt L R + ; L Ω and, for all T >, olds te following d u t + cdu dt dt t, w + D 1 ut, w = D us, w ds + ft, w, a. e. in, T, w H 1 Ω, du dt = u 1, u = u. 6 In 6 te inner products in L Ω and L Ω n are denoted indifferently by.,.. Teir norms will be also represented indifferently by.. Te main objective of tis paper is te analysis of an energy functional under general assumptions and te illustration of te qualitative beaviour of numerical solutions under several different coices for te parameters. We sall prove tat a suitable energy functional converges to zero as t. Numerical wave equations tat mimic teir continuous counterpart will be also considered and teir beaviour will be explored. Te paper is organized as follows. In Section we introduce te new energy functional and we prove tat under convenient assumptions we ave du lim t dt = in L Ω, lim ut = in H 1 Ω, t and lim t K er t s usds = in L Ω. A finite element metod is introduced in Section 3 tat mimics energy beavior of te IBVP studied in tis paper. Te beavior of te IBVP 1, 5 wit omogeneous Diriclet boundary conditions and in Section 5 we summarize some conclusions. Energy beaviour Te energy functional tat we introduce ere extend several definitions introduced before in te literature. For instance, in [4] and [16] te autors considered te classical energy functional Eut = ut + ut, wile in [1] a term was added to te last energy functional induced by te boundary conditions. Also in [3], for a quasilinear problem, a term related wit te reaction term c CMMSE ISBN:

4 Decay of solutions of wave equations wit memory was also added. In [1] te energy functional Eut = 1 du dt K er t sds ut + K er t s ut us ds, 7 was introduced. A similar definition to 7 was considered in [16] but wit te last term of 7 replaced by K er t s ut us ds. In te first result tat we present we establis an estimate for te usual energy for te wave equation Eut = du dt t + ut 1 + us ds ut 8 + us ds, for t >, were u is a solution of 6. Under suitable regularity conditions, it can be sown te following result. Teorem 1. Let u C,, L Ω C 1,, H 1 Ω be a solution of 6 for D 1 > D, K er s = Ke βs, c L Ω satisfying c c > on Ω. 9 and f =. If tere exists a positive constant γ suc tat γ > min { c, β + K}, and } γ β max {γ c, D K { } γ <, 1 min 1, γ γ β K γ c, D 1 D, D, D K ten lim t du dt t + ut t 1 + K er t s us ds ut 11 t + K er t s us ds + e γt s us ds =. 3 Decay decreasing of numerical waves Te study of numerical metods to solve numerically te IBVP 1, 5 wit omogeneous Diriclet boundary conditions was presented for instance in [6], [14], [13], [15] and some of te references of tese papers. c CMMSE ISBN:

5 J. A. Ferreira, P. de Oliveira, G. Pena In tis section we establis tat numerical approximations for te solution of te IBVP1, 5 wit omogeneous Diriclet boundary conditions presents te same qualitative beaviour of te solution of tis problem. Let Ω R be a bounded polygonal domain and let > be a fixed parameter and let T be an admissible triangulation of Ω wit diameter, tat is, = max T diam, were diam denotes te diameter of. Let V be te space of piecewise polynomials of degree m defined in T, tat is V = {v C Ω : v = on Ω, v = p m in, T }, were p m denotes a polynomial of degree at most m. By P Ω and P Ω we represent te set of nodes of T on Ω and Ω, respectively. Let {φ P, P P Ω } be a basis of V. Te finite element approximation for te solution of te IBVP 1, 5 wit omogeneous Diriclet boundary conditions is u x, t = P P Ω α P tφ P x tat satisfies te following d u dt t + cdu dt t, w + u t, w = D du dt = u 1,, u s, w ds + ft, w, a. e. in, T, w V, u = u,. 1 In 1 u 1, and u, are approximations of u 1 and u in V. To compute u t we need to solve te following system of second order integro-differential equations M α t + C α t + A αt = α = U 1,, α = U,, B αs ds + F t, t >, were αt = [α P t P PΩ ], U i,, i =, 1, are te vectors wose components are te coordinates of u i,, i =, 1, wit respect to te basis {φ P, P P Ω }, and M = [φ P, φ Q P,Q PΩ ], C = [cφ P, φ Q P,Q PΩ ], A = [ φ P, φ Q P,Q PΩ ], [ ] D B = φ P, φ Q P,Q PΩ, F t = [ft, φ Q Q PΩ ]. 13 c CMMSE ISBN:

6 Decay of solutions of wave equations wit memory Introducing te new variable Zt = z 1 t, z t were z 1 t = αt, z t = α t, ten te initial value problem 13 of second order is equivalent to t Z t = A Zt + K er t sb Zs ds + F t, t >, 14 Z = U, were [ A = M 1 I A M 1 C ] [ M 1, B = B ] [, F t = M 1 F ] [ U,, U = U 1, ]. As te unique solution of te IVP 14 is smoot enoug, ten for te unique solution u t V of 1 it can be sown te following results: Teorem. Under te assumptions of Teorem 1 we ave du lim t dt t t + u t 1 + K er t s u s ds u t 15 t + K er t s u s ds + e γt s u s ds =. 4 Numerical results In tis section we illustrate te qualitative beaviour of numerical solutions of 13. We now introduce te specifics of our test problems. Given te weak formulation 6, we specify te domain Ω = 1, 1 and te initial data ux, y, = e x +y du.1, x, y, = for dt x, y Ω. Following te spatial discretisation in 1, we introduce te time step t and a uniform partition t j = j t, j =, 1,,..., N = [ T t ]. Applying standard centered finite differences scemes in time and te composite trapezoidal rule to te formulation 13, te following second order in time metod is obtained: u n+1 u n + un 1 = D t t n j=, v + c u n+1 u n 1, v t + D 1 u n+1, v = 16 e t n+1 t j+1 u j+1 + e tn+1 t j u j, v were u j is an approximation for ut j, j =, 1,..., N. c CMMSE ISBN:

7 J. A. Ferreira, P. de Oliveira, G. Pena Let I n+1 = D t I n satisfies n j= I n+1 = e t I 1 = D t e t n+1 t j+1 u j+1 + e tn+1 t j u j. It is straigtforward tat In + D t e t u n + u n+1 e t u + u 1. Wit tis new notation, metod 16 can be rewritten as 1 t + c u n+1 t, v + D 1 D t c = t un + t 1 t u n 1, v, n > 1 u n+1, v + e t In, v. Remark 1. For efficiency reasons, te rigt and side term sould not be computed as it is in formula 16 but rater as recursion formula 17 to avoid te computational cost induced by te sum. Te discrete energy obtained from E,n by applying te same integration scemes as in 16, for a selection of parameters and D. For clarity, te discrete energy is calculated u n as E,n = un + u n t 1 + I n u n n, n. 4.1 Solution s beaviour wit damping In te presence of a positive damping factor c, te numerical solutions tend to zero over time. Tis beaviour is clearly illustrated by Figure??, wic plots te discrete energy function, for different values of D and. It can be observed also tat te larger te damping factor, te faster te energy approximates zero. A similar result is observed wen analysing te numerical solution at te central point, of te square [ 1, 1]. As expected from te previous results, te solution at tis point approximates zero. In Figure?? we plot te numerical solution at tis point, for te same profiles as in Figure??. It is observed tat te smaller te value D is, te closer te solutions are, for different values of, to te solution of te limit case D 1 = 1 and D =. 4. Limiting case as tends to zero For a fixed value of D, te variation of appears to induce a different time scale on te oscillations of te solutions for smaller values of D suc difference is reduced due to te previous conclusions. To furter investigate te beaviour of te numerical solution for varying, we calculated te restriction of te numerical solutions, at time t = 4, in te set c CMMSE ISBN:

8 Decay of solutions of wave equations wit memory [ 1, 1] {}. Tese results are plotted in Figure??. Combining te information from Figures??,?? and?? it seems apparent tat as approaces zero, te corresponding numerical solution approximates te numerical solution obtained taking te pure wave equation wit damping effect included wit wave coefficient D 1 D. In fact, if we consider te differential term It; := D us ds it can be sown tat for a sufficiently smoot function u and fixed t >, lim It; = D ut wic seds some ligt into te observed + beaviour. However, we do not ave, at tis point, a rigorous proof to analytically support tis statement. 5 Conclusions Tis paper establises bounds for an energy function associated wit te solution of a linear wave equation wit a memory term. Under certain conditions, te energy converges to zero as t. It was also sown tat a semi-discrete counterpart of te equation obtained by discretisation in space wit finite elements inerits te same property. Te numerical waves studied also exibit te same convergence to zero of a discretised energy function. It is moreover noticeable tat as te coefficient D approximates zero, te solutions approximate te solution of a pure wave equation wit no memory. Also, as approximates zero, a similar beaviour is observed. Acknowledgements Tis work was partially supported by te Centro de Matemática da Universidade de Coimbra CMUC, funded by te European Regional Development Fund troug te program COMPETE and by te Portuguese Government troug te FCT - Fundação para a Ciência e a Tecnologia under te project PEst-C/MAT/UI34/13. References [1] R. G. C. Almeida, M. L. Santos, Lack of exponential decay of a coupled system of wave equations wit memory, Nonlinear Anal Real World Appl [] J.R. Branco, J.A. Ferreira, A singular perturbation of te eat equation wit memory, J. Comp. Appl. Mat [3] M. M. Cavalcanti, V.N.D. Cavalcanti, J.A. Soriano, Exponential decay for te solution of semilinear viscoelastic wave equations wit localized damping, Electron. J. Diff. Equat c CMMSE ISBN:

9 J. A. Ferreira, P. de Oliveira, G. Pena [4] M. M. Cavalcanti, V.N.D. Cavalcanti,P. Martinez, General decay rate for viscoelastic diisipative systems, Nonlinear Anal. Teor. Met. Appl [5] M. Fabrizio, A. Morro, Matematical Problems in Linear Viscoelasticity, SIAM, Piladelpia, 199. [6] V. Janovský, S. Saw, M.K. Warby, J.R. Witeman, Numerical metods for treating problems of viscoelastic isotropic solid deformation, J. Comput. Appl. Mat [7] M. Kafini, N. Tatar, A decay result to a viscoelastic problem in R n wit an oscillating kernel, J. Mat. Pys [8] S. Messaoudi, M. Mustafa, A general stability result for a quasilinear wave equation wit memory, Nonlinear Anal. R. World Appl [9] J. A. Noel, A nonlinear yperbolic Volterra equation, in Volterra Equations, S-O Londen and O. J. Staffans edts, Lect. Notes Mat [1] J. Y. Park, S. H. Park, Decay rate estimates for wave equations wit memory type wit acoustic boundary conditions, Nonlinear Anal. Teor. Met. Appl [11] M. Renardy, W. Hrusa, J. Noel, Matematical Problems in Viscoelasticity, Longman Scientific and Tecnical, [1] M. L. Santos, Decay rates for solutions of a system of wave equations wit memory, Electron. J. Differ. Equat [13] S. Saw, J. Witeman, Towards adaptive finite element scemes for partial differential Volterra equation solvers, Adv. Comput. Mat [14] S. Saw,J. Witeman, A posteriori error estimates for spacetime finite element approximation of quasistatic ereditary linear viscoelasticity problems, Comput. Met. Appl. Mec. Eng [15] S. Saw, M.K. Warby, J.R. Witeman, Discrete scemes for treating ereditary problems of viscoelasticity and applications, J. Compt. Appl. Mat [16] N. Tatar, Arbitrary decays in linear viscoelasticity, J. Mat. Pys c CMMSE ISBN: