Optimal homotopy asymptotic method for solving of a ...

Optimal homotopy asymptotic method for solving of a nonlinear problem in elasticity R.-D. Ene

V. Marinca

R. Negrea

Department of Mathematics Politehnica University of Timisoara Timisoara, 300006, Romania Email: [email protected]

Department of Politehnica University of Timisoara Timisoara, 300006, Romania Email: [email protected]

Department of Mathematics Politehnica University of Timisoara Timisoara, 300006, Romania Email: [email protected], [email protected]

B. C˘aruntu Department of Mathematics Politehnica University of Timisoara Timisoara, 300006, Romania Email: [email protected] Abstract—In this paper a homotopy approach, called the optimal homotopy asymptotic method (OHAM) is presented as a new and powerful technique for analytical treatment of nonlinear problem related to the stress and deformation state of a thin elastic plate. This technique combines the features of the homotopy concept with an efficient computational algorithm which provides a simple and rigorous procedure to control the convergence of the solution. An excellent agreement is found between the results obtined using OHAM and numerical integration results.

I. I NTRODUCTION Solving nonlinear problems is an important task for scientists. Perturbation methods have been successfully employed to determine approximate solutions to weakly nonlinear problems [1]. There are widely used methods such as the harmonic balance method [1], the Lindstedt-Poincar´ e method [2], the Krylov-Bogoliubov-Mitropolsky method [3], the averaging method [4] or the multiple scales method [5]. Unfortunately some of them are invalid for strongly nonlinear problems and for parameters beyond a certain specified range. That is why new analytical techniques which are valid for strongly nonlinear problems are needed to overcome these shortcomings. In recent years, a growing interest towards the application of homotopy techniques in solving strongly non-linear problems has appeared. In 1992, S. J. Liao employed the basic ideas of the homotopy from topology to propose a general analytic method for nonlinear problems, namely the homotopy analysis method [6]. Different from Liao’s method, Ji-H He proposed in 1998 the homotopy perturbation method [7] which is in fact a new perturbation technique coupled with the homotopy technique. In this paper, a different homotopy approach, namely optimal homotopy asymptotic method [8], [9] is proposed to analytically solve nonlinear problems with particular emphasis on stress and deformation state of a thin elastic plate. The efficiency of our procedure, which does not require a small parameter in the equation, is based on the construction

and determination of the auxiliary functions combined with a convenient way to optimally control the convergence of the solution. The domain of elasticity is in attention of several researchers. Rajagopal K. R. and Wineman A. S. [10] established several new exact solutions to boundary value problems in non-linear elasticity. They show that in the case of the classic torsion solution, there exist an infinity of solutions which are not symmetric. Apostol B.-F. [11] shows that anharmonic corrections to the elastic energy may lead to unphysical solutions for longitudinal deformations such that third-order anharmonic terms is defined as the continuum limit of the Fermi-PastaUlam equation. In the paper [12] Bokhari A. H., Kara A. H. and Zaman F. D. show that a similarity analysis of a nonlinear wave equation in elasticity can be tackled using the group theoretical method. The symmetry transformation give rise to exact solutions via the method of invariants. Mustafa M. T. and Masood K. [13] applied Lie symmetry method to analyze a nonlinear elastic wave equation for longitudinal deformations with third-order anharmonic corrections to the elastic energy. Along with solutions with time-dependent singularities, they also obtain solutions which do not exhibit time-dependent singularities.

II. P RELIMINARY RESULTS

A continuum elastic medium occupies a bounded domain Ω ⊂ R3 , with piecewise smooth boundary. The movement of the medium is described with respect to ortonormat frame R = {O; e1 , e2 , e3 }. The governing equations system of the

stress and deformation state are [14]:  3 X ∂    (tik ) + ρ0 fi∗ = ρ0 u ¨i , x ∈ Ω, t > 0,    ∂x k  k=1     tij = λγrr δij + 2µγij , i, j = 1, 2, 3      3   ∂uj X ∂ur ∂ur  1  ∂ui    , + + · γ = ij  2 ∂xj ∂xi r=1 ∂xi ∂xj

(1)

in which tij represents the elastic stress tensor, γij is the elastic deformation tensor, u = (u1 , u2 , u3 ) is the volume forces field (assumed to be a given continuous function), λ and µ are the ∗ Lam´ e’s elastic constant, ρ0 is the density of medium, f = (f1∗ , f2∗ , f3∗ ) is the volume forces, the symbol ”dot” denotes the first derivative with respect to the time variable t. In the following, we assumed that the Ox1 axis coincides with the propagation direction of the longitudinal elastic wave, i.e.: u1 = u1 (x1 , t), u2 = u2 (x1 , t), u3 = u3 (x1 , t) . (2) f1∗ = f1∗ (x1 , t), f2∗ = f3∗ = 0, x1 ∈ [a, b], t ≥ 0 From the equations (1)1 and the relations (2) we obtain the following systems of equations:   2 ∂u1 ∂ 2 u1   2 ∂ u1  · + + c  1   ∂x21 ∂x1 ∂x21        ∂u ∂ 2 u ∂u3 ∂ 2 u3  2 2 + · · +c21 + f1∗ = u ¨1 . 2 2  ∂x ∂x ∂x ∂x 1 1  1 1       ∂ 2 u3 ∂ 2 u2    c22 =u ¨2 , c22 =u ¨3 2 ∂x1 ∂x21

 ∂2u

1 ∂x21

+

∂u1 ∂ 2 u1  · =u ¨1 . ∂x1 ∂x21

(5)

(6)

For simplicity, we denote u1 (x1 , t) = U (x, t). We write the problem (5) in the form [8]: L(U (x, t)) − N (U (x, t)) = 0,

(7)

where the linear and nonlinear operators are respectively: ¨ (x, t) L(U (x, t)) = U N (U (x, t)) = f1 (x, t) + c21



∂2U ∂x2

+

∂U ∂x

·

∂2U ∂x2

 .

(8)

Using the OHAM method, we construct the following family of equations corresponding to equation (7): (1 − p)L(U (x, t, Kn )) = = H(x, t, p) · [L(U (x, t, Kn )) − N (U (x, t, Kn ))],

(9)

where p is an embedding parameter, p ∈ [0, 1] and the function H can be written in the form: (3)

The equation (3)1 is named the propagation equation of the longitudinal elastic wave (in the Ox1 direction axis) in the presence of the volume forces, taking into account the shear phenomena. In the absence of volume forces and of the shear phenomena, the equation (3)1 is reduced to the following equation: c21

equation (3)1 , which can be written in the form:   2 ∂u1 ∂ 2 u1   2 ∂ u1  + = 0, t > 0, · u ¨ − f − c  1 1 1   ∂x21 ∂x1 ∂x21          u1 (x1 , 0) = v0 (x1 ) , x1 ∈ [a, b]  ∂u1   (x1 , 0) = v1 (x1 )   ∂t        ∂u1 ∂u1   (a, t) = (b, t) = 0, t ≥ 0 ∂x1 ∂x1 where v0 and v1 are given continuous functions and  ∂u ∂ 2 u ∂u3 ∂ 2 u3  2 2 + + f1∗ , f1 = c21 · · ∂x1 ∂x21 ∂x1 ∂x21

(4)

The equation (4) was treat by the several authors [10], [11], [12], [13]. In the following, we treat the general equation (3)1 , which must contain as particular case the equation (4). III. M AIN RESULTS In this section we illustrate the OHAM method for the initial-boundary value problem of the longitudinal elastic wave

H(x, t, p) = ph1 (x, t, Kn1 ) + p2 h2 (x, t, Kn2 ) + ... ,

(10)

hk (x, t, Kni ), ni = 1, 2, ... being the auxiliary functions (control functions) (see [8]) defined below and Kni are unknown constants in this moment. With the OHAM method, we seek the formal solution of the problem (5) of the form: X U (x, t, Kn ) = U0 (x, t) + Ui (x, t, Kn ) · pi , n = 1, 2, ... . i≥1

(11) Substituting the relations (10) and (11) in relation (9) and using the notations (8), we obtain the relation: h i ∂2 (1 − p) ∂t U (x, t) + pU (x, t, K ) + ... = 2 0 1 n   n 2h ∂ ph1 (x, t, Kn1 ) + p2 h2 (x, t, Kn2 ) + ... · ∂t 2 U0 (x, t)+ i 2 +pU1 (x, t, Kn1 ) + p U2 (x, t, Kn2 ) + ... − f1 (x, t)− h 2 ∂ −c21 ∂x U0 (x, t) + pU1 (x, t, Kn1 ) + p2 U2 (x, t, Kn2 )+ 2   ∂ +... + ∂x U0 (x, t) + pU1 (x, t, Kn1 )+   ∂2 p2 U2 (x, t, Kn2 ) + ... · ∂x U0 (x, t) + pU1 (x, t, Kn1 )+ 2 io 2 p U2 (x, t, Kn2 ) + ... (12) for all p ∈ [0, 1].

Identifying coefficients of powers p0 , p1 , p2 , ... from relation (12), we obtain the following linear set of equations: ¨0 (x, t) = 0, U ¨1 (x, t) − U ¨0 (x, t) = U i h  2 2 ¨0 − f1 − c21 ∂ U0 + ∂U0 · ∂ U0 , h1 (x, t, Kn1 ) U ∂x2 ∂x ∂x2 ¨2 (x, t) − U ¨1 (x, t) = U h  2 2 ¨1 − c21 ∂ U1 + ∂U0 · ∂ U1 + h1 (x, t, Kn1 ) U 2 ∂x ∂x2 i h∂x ∂U1 ∂ 2 U0 ¨ + ∂x · ∂x2 + +h2 (x, t, Kn2 ) U0 − f1 − i  ∂U0 ∂ 2 U0 2 ∂ 2 U0 −c1 ∂x2 + ∂x · ∂x2 , ... (13) where hi (x, t, Kj ) are arbitrary functions. For the set of equations (13) we must take the following initialboundary conditions: U0 (x, 0) = v0 (x)

∂U0 ∂x (a, t)

=0

∂U0 ∂t (x, 0)

∂U0 ∂x (b, t)

=0

Un (x, 0) = 0

∂Un ∂x (a, t)

=0

∂Un ∂t (x, 0)

∂Un ∂x (b, t)

=0

= v1 (x)

(14) , n = 1, 2, ... =0

Finally, the solutions U0 , U1 , ... of the initial-boundary value problems (13)-(14) allows us to determine the formal solution (11). The m-order approximate solution of the problem (5) is obtained truncating equation (11): U (x, t, Kn ) = U0 (x, t) +

m X

Ui (x, t, Kn ), n = 1, 2, ... .

i=1

(15) Introducing the relation (15) into the relation (7), the approximating rest is written as: R(x, t, Kn ) = L(U (x, t, Kn )) − N (U (x, t, Kn )).

(16)

The minimum conditions of the approximating rest, allows to determine the control constants Kn , n = 1, 2, .... For example, using the least-squares method, the rest functional Z bZ T  J(Kn ) = R2 (x, t, Kn ) dt dx (17) a

IV. N UMERICAL RESULTS Now, using OHAM technique, we solve the initial-boundary value problem (3) for the longitudinal deformation of a thin elastic plate, taking into account the shear phenomena. We suppose that under the action of an volume force: f1∗ (x, t) = cos(4πx) sin(4πc2 t)

(19)

for x ∈ [0, 1], t ∈ [0, 1] (a = 0, b = 1, T = 1). The boundary plate is assumed to be free of stress, i.e. such that we have  ∂u1   (0, t) =   ∂x      ∂u2 (0, t) =  ∂x         ∂u3 (0, t) = ∂x

∂u1 (1, t) = 0 ∂x ∂u2 (1, t) = 0 , ∂x

t≥0.

(20)

∂u3 (1, t) = 0 ∂x

The equation (20) are known as the Neumann’s type homogeneous boundary conditions. We assume that the initial conditions are:    u1 (x, 0) = cos(2πx),        u2 (x, 0) = cos(4πx),          u3 (x, 0) = cos(4πx),

∂u1 (x, 0) = 0, ∂t ∂u2 (x, 0) = 0, ∂t

,

x ∈ [0, 1].

∂u3 (x, 0) = 0, ∂t

(21) The solutions of the problems (3)2 -(21)2 -(20)2 and (3)3 (21)3 -(20)3 are: u2 (x, t) = cos(4πx) cos(4πc2 t), u3 (x, t) = cos(4πx) cos(4πc2 t)

(22)

respectively. For the nonlinear problem (3)1 -(21)1 -(20)1 , using OHAM technique, we can construct the associated family of equations:

0

is minimum iff the relations ∂J ∂J = = ... = 0 ∂K1 ∂K2

(18)

are holds, where T is the boundary of the domain for variable t. The relations (18) lead us to the nonlinear algebraic equations system, with unknown Kn , n = 1, 2, ... which can be solved using a well-know numerically method. Thus, we introducing the convergence control constants Kn , n = 1, 2, ... into relation (15), we obtain the m-order approximate solution for the problem (5).

h  2 2 i ¨ = ph1 (x, t, Kn )· U ¨ −f1 −c21 ∂ U + ∂U · ∂ U . (1−p)U 1 ∂x2 ∂x ∂x2 (23) Following [8], for equation (23) we suppose that the solution of the form: U (x, t) = U0 (x, t) + pU1 (x, t) + p2 U2 (x, t) + ...

(24)

with p ∈ [0, 1]. Substituting the relation (24) into relation (23), taking into account the initial conditions (21) and boundary conditions (20), where by identification, we get the problem:

The approximate rest is:  ¨0 = 0  U      U0 (x, 0) = cos(2πx),

∂U0 (x, 0) = 0, ∂t

      ∂U0 (0, t) = ∂U0 (1, t) = 0, ∂x ∂x

x ∈ [0, 1]

t≥0 (25)

with solution

 2 2  ¨ − c2 ∂ U + ∂U · ∂ U − f (x, t). R(x, t, K1 , K2 , K3 ) = U 1 1 ∂x2 ∂x ∂x2 (32) Using the least-squares method, the rest functional Z 1Z 1  R2 (x, t, K1 , K2 , K3 ) dt dx J(K1 , K2 , K3 ) = 0

U0 (x, t) = cos(2πx)

(26)

0

∂J ∂J ∂J = = = 0. ∂K1 ∂K2 ∂K3

and the problem    2 2  ¨1 = h1 (x, t, Kn )[U ¨0 − c21 ∂ U0 + ∂U0 · ∂ U0 − f1 ],  U  1 2 2  ∂x ∂x ∂x      ∂U1 U1 (x, 0) = 0, (x, 0) = 0, x ∈ [0, 1]  ∂t         ∂U1 (0, t) = ∂U1 (1, t) = 0, t ≥ 0 ∂x ∂x (27) respectively. For the problem (27), we choose the auxiliary function h1 of the form:

(33)

is minimal, if (34)

The numerical solution of the nonlinear algebraic equations system (34) (using Mathematica soft) is: K1 = −0.974115, K2 = 0.00903759, K3 = 0.00251359.

(35)

The comparison between the numerical solution using OHAM technique and the numerical solution using the Mathematica soft, (for several values of time variable t) is presented in the tables below: Table 1. Numerical data for the time moment t = 4/20 error uOHAM unumeric |uOHAM − unumeric | 0 1.01148 1.0154 0.00391356 0.1 0.818067 0.818443 0.000375917 0.2 0.292274 0.291652 0.000622855 0.3 -0.311349 -0.311393 0.0000431229 0.4 -0.810781 -0.810854 0.0000726081 0.5 -0.987906 -0.987477 0.000428775 0.6 -0.799473 -0.799193 0.000280121 0.7 -0.325571 -0.326062 0.000491153 0.8 0.306496 0.306352 0.00014402 0.9 0.806759 0.806675 0.0000846761 1 1.01148 1.00827 0.00321426 x

h1 (x, t, K1 , K2 , K3 ) = K1 + K2 cos(2πc2 t) + K3 cos(6πc2 t). (28) The function (28) is not unique. Also we can choose the auxiliary function by of the forms: h∗1 (x, t, K1 , K2 , K3 , K4 ) = K1 + K2 cos(2πc2 t)+ +K3 cos(6πc2 t) + K4 cos(10πc2 t)

(29)

e h1 (x, t, K1 , K2 , K3 , K4 ) = K1 + K2 cos(2πc2 t) and so on. Using the relations (28), (26), (22) and (19), the problem (27) has the solution:

Table 1. Numerical data for the time moment t = 10/20 error uOHAM unumeric |uOHAM − unumeric | 0 1.03644 1.05778 0.0213382 0.1 0.853316 0.856037 0.00272153 0.2 0.235184 0.231184 0.00399924 0.3 -0.29725 -0.296251 0.000999328 0.4 -0.829608 -0.830496 0.000887371 0.5 -0.959725 -0.957916 0.00180935 0.6 -0.761619 -0.761468 0.000151483 0.7 -0.381667 -0.381752 0.000085842 0.8 0.3196 0.318194 0.00140588 0.9 0.785326 0.785282 0.0000445205 1 1.03644 1.01542 0.0210204 x

U1 (x, t) = 0.0004K2 cos(2πx) + 0.00789568K1 t2 cos(2πx) −0.0795775K1 t cos(4πx) − 0.106103K2 t cos(4πx) +0.063662K3 t cos(4πx) + 0.0126651K2 cos(4πx) sin(2πt) −0.0126651K3 cos(4πx) sin(2πt) +0.00633257K1 cos(4πx) sin(4πt) +0.00140724K2 cos(4πx) sin(6πt) − 0.00125664K2 sin(4πx)− −0.024805K1 t2 sin(4πx) + 0.00125664K2 cos(2πt) sin(4πx) −0.0108127K2 sin(8πx) − 0.00624614K3 sin(8πx) −0.19844K1 t2 sin(8πx) + 0.0100531K2 cos(2πt) sin(8πx) +0.00502655K3 cos(2πt) sin(8πx) +0.00111701K3 cos(6πt) sin(8πx) These numerical data can be viewed in the figures 1 and 2: (30) In this way, the first-order approximate solution of the problem (3)1 -(21)1 -(20)1 is written as (for p = 1): From this numerical comparison, we deduce the successfully implementation of the OHAM technique used to numerU (x, t, K1 , K2 , K3 ) = U0 (x, t)+U1 (x, t, K1 , K2 , K3 ). (31) ically solve the nonlinear partial differential equations.

Table 1. Numerical data for the time moment t = 16/20 error x uOHAM unumeric |uOHAM − unumeric | 0 1.06001 1.10698 0.0469745 0.1 0.912716 0.923667 0.0109508 0.2 0.146073 0.131665 0.0144081 0.3 -0.251131 -0.244284 0.00684743 0.4 -0.872587 -0.877519 0.0049318 0.5 -0.930151 -0.925629 0.0045213 0.6 -0.697358 -0.701021 0.00366341 0.7 -0.468921 -0.460496 0.00842449 0.8 0.363862 0.354805 0.00905773 0.9 0.737487 0.74028 0.00279302 1 1.06001 1.00223 0.0577795

1.0

0.5

0.2

0.4

0.6

0.8

1.0

0.6

0.8

1.0

-0.5

-1.0 1.0

Fig. 3. 1.0

0.5

0.5 0.2

0.4

0.6

0.8

1.0

-0.5

0.2

0.4

-0.5

-1.0

Fig. 1.

a) Numerical solution using Mathematica soft -1.0

1.0

Fig. 4.

a) Numerical solution using Mathematica soft

1.0

0.5

0.5 0.2

0.4

0.6

0.8

1.0

-0.5

0.2

0.4

0.6

0.8

1.0

-0.5

-1.0

Fig. 2.

b) Numerical solution using OHAM -1.0

V. C ONCLUSION In this paper OHAM is employed to propose a new analytic approximate solution for the nonlinear problem of stress and deformation state of a thin elastic plate. The proposed procedure is valid even if the nonlinear equation does not contain any small parameter. OHAM provides us with a simple and rigorous way to control and adjust the convergence of the solution trough the auxiliary function hi (x, t, p) involving several constants Ki which are optimally determined. In the proposed procedure, iterations are performed in a very simple manner by identifying some coefficients, and therefore, very good approximations are obtained in one term. Actually, the

Fig. 5.

b) Numerical solution using OHAM

capital strength of the proposed procedure is its fast convergence, since after only one iteration it converges to the exact solution, which proves that this method is very effective in practice. R EFERENCES [1] A. H. Nayfeh, Introduction to perturbation techniques, John Wiley, N. Y., 1981. [2] , L. Cveticanin, I. Kovacic, Parametrically excited vibrations of a oscillator with strong cubic negative nonlinearity, Journal of Sound and Vibration, 304, 2007, 201–212.

1.0

1.0

0.5

0.5

0.2

0.4

0.6

0.8

1.0

0.2

-0.5

-0.5

-1.0

-1.0

1.0

0.5

0.4

0.6

0.8

1.0

-0.5

-1.0

Fig. 7.

a) Numerical solution using Mathematica soft

1.0

0.5

0.2

0.4

0.6

0.8

1.0

-0.5

-1.0

Fig. 8.

0.6

0.8

1.0

Fig. 9.

Fig. 6.

0.2

0.4

b) Numerical solution using OHAM

[3] N. N. Bogoliubov, Y. A. Mitropolsky, Asymptotic methods in the theory of nonlinear oscillations, Gordon and Breach, N. Y., 1961. [4] J. A. Sanders, F. Verhulst, Averaging methods in nonlinear dynamical systems, Springer-Verlag, N. Y., 1985. [5] M. P. Cartmell, S. W. Ziegler, Khanin, R., Forehand, D. I. M., Multiple scales analyses of the dynamics of weakly nonlinear mechanical systems, Appl. Mech. Rev., 56, 2003, 453–493. [6] S. J. Liao, A second-order approximate analytical solution of a simple pendulum by the process analysis method, ASME, J. Appl. Mech., 59, 1992, 970–975. [7] J. H. He, An approximate solution technique depending upon an artificial parameter, Commun Nonlin. Sci. Numer. Simul., 3, 1998, 92–97. [8] V. Marinca, N. Heris¸anu, Nonlinear Dynamical Systems in Engineering

- Some Approximate Approaches, Springer Verlag, Heidelberg, 2011. [9] V. Marinca, N. Heris¸anu, An Optimal Homotopy Asymptotic Approach Applied to Nonlinear MHD Jeffery-Hamel-Flow, Mathematical Problems in Engineering, article ID 169056 (2011) doi: 10.1155/2011/169056. [10] K. R. Rajagopal, A. S. Wineman, New exact solutions in non-linear elasticity, Int. J.. Engng. Sci., 23 (2), 1985, 217–234. [11] B. F. Apostol, On a non-linear wave equation in elasticity, Physics Letters A 318, 2003, 545–552. [12] A. H. Bokhari, A. H. Kara, F. D. Zaman, Exact solutions of some general nonlinear wave equations in elasticity, Nonlinear Dyn., DOI10.2007/s11071-006-9050-z, 6 pages. [13] M. T. Mustafa, K. Masood, Symmetry solutions of a nonlinear elastic wave equation with third-order anharmonic corrections, Appl. Math. Mech. - Engl.Ed., 30 (8), 2009, 1017–1026. ˇ Miroslav, The Mechanics and Thermodynamics of Continuous Media [14] S. (Theoretical and Mathematical Physics), Springer, 1 edition, June 21, 2002.