Numerical method, existence and uniqueness for ... - SciELO

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Volume 24, N. 3, pp. 439–460, 2005 Copyright © 2005 SBMAC ISSN 0101-8205 www.scielo.br/cam

Numerical method, existence and uniqueness for thermoelasticity system with moving boundary M.A. RINCON1 , B.S. SANTOS2 and J. LÍMACO2 1 Instituto de Matemática, Universidade Federal do Rio de Janeiro 2 Instituto de Matemática, Universidade Federal Fluminense

Rio de Janeiro, Brazil E-mails: [email protected] / [email protected] / [email protected]

Abstract. In this work, we are interested in obtaining existence, uniqueness of the solution and an approximate numerical solution for the model of linear thermoelasticity with moving boundary. We apply finite element method with finite difference for evolution in time to obtain an approximate numerical solution. Some numerical experiments were presented to show the moving boundary’s effects for problems in linear thermoelasticity.

Mathematical subject classification: 35A05, 35A40, 65M60, 65M06. Key words: thermoelasticity system; moving boundary; finite element method; finite difference method.

1 Introduction  Let Q t = (x, t) ∈ R2 ; α(t) < x < β(t), 0 < t < T be the non-cylindrical domain with boundary [ {α(t), β(t)} × {t} 6t = 0 0.

where γ (t) = β(t) − α(t),
2

,

for 0 ≤ t ≤ T and 0 ≤ y ≤ 1.

We will now consider a change of variables to transform the domain Q t into a cylindrical domain Q. Observe that, when (x, t) varies in Q t the point (y, t) of R2 , with y = (x − α(t))/γ (t) varies in the cylinder Q = (0, 1) × (0, T ). Thus, we define the application T : Q t → Q = (0, 1) × (0, T )

(x, t) 7→ (y, t) =

 x − α(t)  ,t . γ (t)

(1)

The application T belongs to C 2 and its inverse T −1 is also C 2 . The transformation of a moving boundary domain to a domain with fixed boundary has been employed elsewhere (see [2, 10, 11]). Comp. Appl. Math., Vol. 24, N. 3, 2005

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Doing the change of variable v(y, t) = u(α(t) + γ (t)y, t) and φ(y, t) = θ (α(t) + γ (t)y, t) and applying to the problema (I), we obtain the following equivalent problem defined in a fixed cylindrical domain:  2 ∂ v ∂  ∂v  ∂φ  ∂t 2 − ∂ y a1 (y, t) ∂ y + a2 (t) ∂ y   2  +a3 (y, t) ∂∂y∂tv + a4 (y, t) ∂v = 0, in Q  ∂y    ∂φ ∂ 2φ ∂ 2v ∂φ  − b (t) + b (t) + b3 (y, t) 1 2  ∂t 2 ∂y ∂ y∂t ∂y  (II)  2  +b4 (t) ∂v + b5 (y, t) ∂∂ yv2 = 0, in Q ∂y     v = φ = 0; ∀ (y, t) ∈ 6,   ∂v  v(y, 0) = v0 (y), (y, 0) = v1 (y),  ∂t  φ(y, 0) = φ0 (y),

for

0 < y < 1.

where

b1 (t) = k/γ (t)2 ,

b2 (t) = η2 /γ (t) ,

b3 (y, t) = −(α 0 (t) + γ 0 (t)y)/γ (t) ,

b4 (t) = −γ 0 (t)/γ (t)2 ,

b5 (y, t) = b3 (y, t)/γ (t) ,

a2 (t) = η1 /γ (t) ,

a3 (y, t) = 2b3 (y, t) ,

 2 a1 (y, t) = 1/γ (t)2 − b3 (y, t) ,

a4 (y, t) = −(α 00 (t) + γ 00 (t)y)/γ (t).

Let (( , )), k ∙ k and ( , ), | ∙ |, be respectively the scalar product and the norms in H01 (0, 1) and L 2 (0, 1). We denote by a1 (t, v, w) and b1 (t, v, w) the bilinear forms, continuous, symmetric and coercive, defined in H01 (0, 1) by a1 (t, v, w) =

Z

b1 (t, v, w) =

Z

1

a1 (y, t) 0

∂v ∂w dy, ∂y ∂y (2)

1 0

∂v ∂w b1 (t) dy. ∂y ∂y

2 Existence and uniqueness We shall first establish the existence and uniqueness of problem (II) in Theorem 2 as auxiliary and then prove the following Theorem 1 of the original problem (I). Comp. Appl. Math., Vol. 24, N. 3, 2005

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THERMOELASTICITY SYSTEM WITH MOVING BOUNDARY

Theorem 1. data

Under the hypotheses (H1), (H2) and (H3) and given the initial {u 0 , θ0 } ∈ H01 (0 ) ∩ H 2 (0 ),

u 1 ∈ H01 (0 ),

there exist functions {u; θ } : Q t → R, solution of Problem (I) in Q t , satisfying the following conditions: 1. u ∈ L ∞ (0, T ; H01 (t ) ∩ H 2 (t )), u 00 ∈ L ∞ (0, T ; L 2 (t )),

u 0 ∈ L ∞ (0, T ; H01 (t )),

2. θ ∈ L 2 (0, T ; H01 (t ) ∩ H 2 (t )), θ 0 ∈ L 2 (0, T ; H01 (t )). Theorem 2. data

Under the hypotheses (H1), (H2) and (H3) and given the initial {v0 , φ0 } ∈ H01 (0, 1) ∩ H 2 (0, 1),

v1 ∈ H01 (0, 1),

there exists functions {v; φ} : Q → R, solution of Problem (II) in Q, satisfying the following conditions: 1. v ∈ L ∞ (0, T ; H01 (0, 1) ∩ H 2 (0, 1)), v 0 ∈ L ∞ (0, T ; H01 (0, 1)), v 00 ∈ L ∞ (0, T ; L 2 (0, 1)), 2. φ ∈ L 2 (0, T ; H01 (0, 1)) ∩ H 2 (0, 1), φ 0 ∈ L 2 (0, T ; H01 (0, 1)). Proof of Theorem 2. To prove the theorem, we introduce the approximate solutions. Let T > 0 and denote by Vm the subspace spanned by {w1 , w2 , ..., wm }, where {wν , λν ; ν = 1, ∙ ∙ ∙ m} are solutions of the spectral problem ((wi , v)) = μ(wi , v), ∀v ∈ H01 (0, 1). If {vm ; φm } ∈ Vm then it can be represented by vm =

m X

dνm (t)wν (y),

ν=1

Comp. Appl. Math., Vol. 24, N. 3, 2005

φm =

m X ν=1

gνm (t)wν (y).

(3)

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Let us consider {vm ; φm } solutions of the system of ordinary differential equations,    ∂φm 00 , w) + a (t, v , w) + a , w (v 1 m 2  m ∂y       0 ∂vm ∂vm   + a3 , w + a4 , w = 0,  ∂y ∂y   0      (φ 0 , w) + b (t, φ , w) + b ∂vm , w + b ∂φm , w  m 1 m 2 3 ∂y ∂y       (III)  ∂w ∂v ∂v m m  + 2b4 , w + b5 , = 0,  ∂ y ∂ y ∂y    vm (0) = v0m → v0 , in H 1 (0, 1) ∩ H 2 (0, 1), 0    0  vm (0) = v1m → v1 in H01 (0, 1),   φm (0) = φ0m → φ0 in H01 (0, 1) ∩ H 2 (0, 1),

where w ∈ Vm . The system (III) has local solution in the interval (0, Tm ). To extend the local solution to the interval (0, T ) independent of m, the following estimates are necessary: A priori estimate Taking w = vm0 and w = φm in the equation (III)1 and (III)2 , respectively, we get  ∂φ  1 d 0 2 m |vm | + a1 (t, vm , vm0 ) + a2 , vm0 2 dt ∂y (4)  ∂v 0   ∂v  m m 0 0 + a3 ,v ,v + a4 = 0, ∂y m ∂y m  ∂v 0   ∂φ  1 d m m |φm |2 + b1 (t, φm , φm ) + b2 , φm + b3 , φm 2 dt ∂y ∂y  ∂v   ∂v ∂φ  m m m + 2b4 , φm + b5 , = 0. ∂y ∂y ∂y

Comp. Appl. Math., Vol. 24, N. 3, 2005

(5)

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THERMOELASTICITY SYSTEM WITH MOVING BOUNDARY

Note that, we have the following relations: 1 d 1  ∂vm ∂w  a1 (t, vm , vm ) − a10 , , 2 dt 2 ∂y ∂y

a1 (t, vm , vm0 ) = a2

 ∂φ

m

∂y

, vm0



= −

 η1  ∂vm0 b2 φm , η2 ∂y

  ∂v 0 γ0 a3 m , vm0 = − |vm0 |2 , ∂y γ



b3

(6)

 ∂φm γ0 |φm |2 , , φm = ∂y 2γ

 ∂v ∂φ  1 m m , b5 ≤ ckvm k2 + b1 (t, φm , φm ). ∂y ∂y 2

Multiplying (4) by (η2 /η1 ), adding it to (5) and using (6) we have  η1 η2 d  0 2 |vm | + a1 (t, vm , vm ) + |φm |2 + b1 (t, φm , φm ) 2η1 dt η2   ≤ C |vm0 |2 + kv0m k2 + |φm |2 .

(7)

Knowing that a1 (t, v, w) and b1 (t, v, w) are coercive forms, by integrating (7) and applying the Gronwall’s inequality, we get |vm0 |2 + kvm k2 + |φm |2 +

Z

t 0

  kφm k2 ≤ c1 |v1m |2 + kv0m k2 + |φ0m |2 ec2 T . (8)

Second estimate Taking the derivative with respect to t, of approximate system (III)1,2 , and also w = vm00 , w = φm0 , respectively, we obtain  ∂φ 0   ∂v 00  m (vm000 , vm00 ) + a1 (t, vm0 , vm00 ) + a2 , vm00 + a3 m , vm00 ∂y ∂y   ∂φ   ∂v 0 m (9) , vm00 + (a30 + a4 ) m , vm00 + a10 (t, vm , vm00 ) + a20 ∂y ∂y  ∂v  m + a40 , vm00 = 0 ∂y Comp. Appl. Math., Vol. 24, N. 3, 2005

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and   ∂φ 0  , φm0 + b3 m , φm0 ∂y ∂y  ∂v 0   ∂v 0 ∂φ 0  m , φm0 − b5 m , m + (2b4 + b20 ) ∂y ∂y ∂y   ∂v   ∂φ m m + b10 (t, φm , φm0 ) + b30 , φm0 + 2b40 , φm0 ∂y ∂y  ∂v ∂φ 0  m + b50 , m = 0. ∂y ∂y

ds(φm00 , φm0 ) + b1 (t, φm0 , φm0 ) + b2

 ∂v 00

m

(10)

We also have the following relations: 0 00 a1 (t, vm , vm )=

1 d 0 1 0 0 0 0 a1 (t, vm , vm ) − a10 (t, vm , vm ), 2 dt 2

 ∂v 00  γ0 00 00 2 a 3 m , vm | , = |vm ∂y γ  ∂v ∂v 0   ∂v 0 ∂v 0  0  d  0 ∂vm ∂vm m m 00 a10 (t, vm , vm , − a100 , m − a10 , m , )= a1 ∂y ∂y ∂y ∂y ∂y ∂y dt  ∂φ 0   00 η1 .b2  ∂vm m 00 a2 =− , vm , φm0 , ∂y η2 ∂y  ∂φ 0  1 γ0 m b3 , φm0 = |φm0 |2 ∂y 2γ

(11)

 0  0 ∂vm ∂vm η2 2 0 0 a 1 ∂ y , ∂ y ≤ Ckvm k + 4η a1 (t, vm , vm ). 1

Multiplying (9) by (η1 /η2 ), adding it to (10) and using (11), we obtain    ∂v ∂v 0  α η2 d m m 00 2 0 0 0 0 |vm | + a1 (t, vm , vm ) + a1 , + |φm | 2η1 dt ∂y ∂y β +

b1 (t, φm0 , φm0 )

≤ C kvm k + 2

kvm0 k2

+

|vm00 |2

+ kφm k + 2

|φm0 |2




(12) .

From (III)1,5 , |vm00 (0)|2 and |φm0 (0)|2 are bounded. Hence, by integrating (12) with respect t and applying the Gronwall’s inequality, we get Z t 0 2 00 2 0 2 kvm k + |vm | + |φm | + kφm0 k2 ≤ C. (13) 0

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THERMOELASTICITY SYSTEM WITH MOVING BOUNDARY

Third estimate Taking w = − ∂ 2 vm /∂ y 2 and w = − ∂ 2 φm /∂ y 2 , in the approximate system (III)1,2 , we have 

vm00 , −

  ∂φ ∂ 2 vm  ∂ 2 vm  ∂ 2 vm  m + a t, v , − + a , − 1 m 2 ∂ y2 ∂ y2 ∂y ∂ y2  ∂v 0 ∂ 2 vm   ∂vm ∂ 2 vm  = 0 ,− 2 + a3 m , − 2 + a4 ∂y ∂y ∂y ∂y

(14)

and 

φm0 , −

  ∂v 0 ∂ 2 φm  ∂ 2 φm  ∂ 2 φm  m + + b t, φ , − b , − 1 m 2 ∂ y2 ∂ y2 ∂y ∂ y2 + 2b4

 ∂v

∂ φm   ∂vm ∂ 3 φm  m , − 2 + b5 ,− 3 = 0. ∂y ∂y ∂y ∂y

(15)

2

Note that, we have the following equalities:   ∂v ∂v   ∂a ∂ 2 vm  m m 1 , + a1 t, vm , − 2 = a1 t, ∂y ∂y ∂y ∂y     2 ∂ φm ∂φm ∂φm b1 t, φm , − 2 = b1 t, , ∂y ∂y ∂y  ∂v ∂ 3 φm   ∂ 2 vm ∂ 2 φm   ∂b5 m b5 , − 3 = b5 2 , − ∂y ∂y ∂y ∂ y2 ∂y

∂vm ∂ 2 vm  , ∂ y ∂ y2 (16) ∂vm ∂ 2 φm  , . ∂ y ∂ y2

From (14), (15) and (16) and since a1 (t, v, w) and b1 (t, v, w) are coercive forms, we obtain ∂ 2 v 2
m (17) 2 ≤ c6 kφm k2 + |vm00 |2 + kvm0 k2 + kvm k2 , ∂y ∂ 2 φ 2
m (18) 2 ≤ c7 |φm0 |2 + kφm k2 + |vm00 |2 + kvm0 k2 + kvm k2 . ∂y The estimates obtained in (8), (13), (17) and (18), permit us to pass the limits in the approximate system (III)1,2 in the Galerkin method and hence, we have proved the existence of solutions {v, φ} in the sense defined in Theorem 2. Comp. Appl. Math., Vol. 24, N. 3, 2005

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Uniqueness of solution ˆ and {e e} be two solutions of Problem (II). Then v = vˆ − e Let {v, ˆ φ} v, φ v and ˆ e φ = φ − φ are also solutions of Problem (II), with null initial conditions. Then, multiplying the equation (II)1,2 , respectively by (η1 /η2 )v and φ, we obtain Z t  0 2 2 2 |v | + kvk + |φ| ≤ c |v 0 |2 + kvk2 + |φ|2 . (19) 0

From Gronwall Lemma, we have |v 0 |2 + kvk2 + |φ|2 = 0 and therefore, we conclude that v = φ = 0 for all 0 < t < T . This completes the proof of Theorem 2.  The original problem (I) Now let us restate the previous results for the original problem (I) in order to prove Theorem 1.

Proof of Theorem 1. Let {v, φ} be a solution of Problem (II), with initial data given by     v0 (y) = u 0 α(0) + γ (0)y , φ0 (y) = θ0 α(0) + γ (0)y ,       v1 (y) = u 1 α(0) + γ (0)y + α 0 (0) + γ 0 (0)y u 00 α(0) + γ (0)y .

Consider the functions u(x, t) = v(y, t) and θ(x, t) = φ(y, t), where x = α(t) + γ (t)y. To verify that u(x, t) and θ(x, t), under the hypotheses of Theorem 1, are a solution of problem (I), it is sufficient to observe that the mapping: (x, t) → ((x − α)/γ , t) of the domain Q t into Q = (0, 1) × (0, T ) is of class C 2 . Since that ∂u 2 1 ∂v 2 ∂θ ∂θ 1. = , η1 = a2 , 2 2 ∂x γ ∂y ∂x ∂y   2 ∂ ∂v ∂ v ∂v 1 ∂ 2v + a3 2. u 00 = v 00 − a1 + a4 + 2 2 ∂y ∂y ∂ y∂t ∂y γ ∂y 2 2 ∂ θ ∂ φ ∂θ ∂φ ∂φ 3. k 2 = b1 2 , = + b3 , ∂x ∂y ∂t ∂t ∂y 2 2 ∂ u ∂ v ∂v 4. η2 = b2 + b4 , ∂ x∂t ∂ y∂t ∂y Comp. Appl. Math., Vol. 24, N. 3, 2005

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and from problem (II) we also have that {u, θ } satisfies the problem (I). The regularity of {v(y, t), φ(y, t)} given by Theorem 2 implies that {u(x, t), θ (x, t)} is a solution of problem (I) and the uniqueness of the solution of problem  (I) is a direct consequence of the uniqueness of problem (II). 3 Approximate solution Our goal in this section is the numerical implementation of approximate solutions. To obtain the numerical approximate solutions we will use both finite element method and finite difference method. Moreover, some numerical experiments will be presented to analyze the effect of the moving boundary in the thermoelasticity system. For convenience, our numerical analysis using finite element method approximation will be based on the equivalent problem (II) in the rectangular domain, instead of the problem (I), for which the domain depends on time. We will consider, in numerical simulations, the case in which the following change in the boundary functions, α(t) = −K (t) and β(t) = K (t), is assumed. Note that, now we have  Q t = (x, t) ∈ R2 ; x = K (t)y, y ∈ (−1, 1), t ∈ (0, T )

(20)

being the non-cylindrical domain with boundary [ 6t = {−K (t), K (t)} × {t}, 0