Commun Nonlinear Sci Numer Simulat 65 (2018) 54–69

Contents lists available at ScienceDirect

Commun Nonlinear Sci Numer Simulat

journal homepage: www.elsevier.com/locate/cnsns

Research paper

Bifurcation & chaos in nonlinear structural dynamics: Novel &

highly efficient optimal-feedback accelerated Picard iteration

algorithms

Xuechuan Wang

∗, Weicheng Pei , Satya N. Atluri

Center for Advanced Research in the Engineering Sciences, Texas Tech University, Lubbock, TX 79415, USA

a r t i c l e i n f o

Article history:

Received 23 January 2018

Revised 11 May 2018

Accepted 14 May 2018

Available online 17 May 2018

Keywords:

Variational method

Picard iteration method

Collocation method

Structural vibrations

Nonlinear dynamics

a b s t r a c t

A new class of algorithms for solving nonlinear structural dynamical problems are de-

rived in the present paper, as being based on optimal-feedback-accelerated Picard itera-

tion, wherein the solution vectors for the displacements and velocities at any time t in a

finitely large time interval t i ≤ t ≤ t i +1 are corrected by a weighted (with a matrix λ) in-

tegral of the error from t i to t . We present 3 approximations to solve the Euler-Lagrange

equations for the optimal weighting functions λ; thus we present 3 algorithms denoted

as Optimal-Feedback-Accelerated Picard Iteration (OFAPI) algorithms-1, 2, 3. The interval

( t i +1 − t i ) in the 3 OFAPI algorithms can be several hundred times larger than the incre-

ment ( �t ) required in the finite difference based implicit or explicit methods, for the same

stability and accuracy. Moreover, the OFAPI algorithms-2, 3 do not require the inversion of

the tangent stiffness matrix, as is required in finite difference based implicit methods. It is

found that OFAPI algorithms-1, 2, 3 (especially OFAPI algorithm-2) require several orders

of magnitude of less computational time than the currently popular implicit and explicit

finite difference methods, and provide better accuracy and convergence.

© 2018 Elsevier B.V. All rights reserved.

1. Introduction

Nonlinearities in the dynamics of structures could arise from various sources such as large deformations, large rotations,

nonlinearities of the material, damping, and boundary conditions, etc. [1] They are so common in real life that the non-

linear behaviors can be found everywhere ranging from civil engineering, automobile engineering, aerospace engineering,

to robotic dynamics, etc. With the development of lighter, more flexible and multifunctional materials and structures, the

effect of nonlinearity will become more significant in these areas. Linearization was once widely used to simplify nonlinear

problems, but it was also realized that the basic principles of linear analysis are not valid even in some cases of weak non-

linearity. The nonlinear dynamical system is characterized by some unique and complex phenomena, including bifurcation,

jump phenomena, snap-through, and chaos, that the linear system cannot exhibit [2] . Analytically solving the problems of

nonlinear dynamical systems is difficult, although some basically qualitative analysis can still be made using perturbation

methods or homotopy analysis methods [3] . In the current state of mathematics, for strongly nonlinear dynamical systems

with complex forms of nonlinearities, the general solutions can only be obtained using numerical integration methods. The

investigation of structural vibration helps to enhance the reliability and performance of structures, as well as revealing

∗ Corresponding author.

E-mail address: xc.wang@ttu.edu (X. Wang).

https://doi.org/10.1016/j.cnsns.2018.05.008

1007-5704/© 2018 Elsevier B.V. All rights reserved.

X. Wang et al. / Commun Nonlinear Sci Numer Simulat 65 (2018) 54–69 55

potential failures. With proper structural models, one can use numerical simulation as an alternative to experiment, and

thus save a lot of time and expense.

In literature, the methods for numerically solving a semi-discrete system of nonlinear ordinary differential equations

in time (after the spatial discretization has been carried out) in structural dynamics, are roughly divided into (i) weak

form methods based on weighted residual concepts and (ii) finite difference methods. Various weak form methods are

developed by using different trial and test functions in the weighted residual formulation in the time domain. Dowell et al.

introduced the Harmonic Balance (HB) method [4] using harmonics as basis functions to approximate both the trial and

test functions, in the Galerkin weak form solution of the semi-discrete nonlinear differential equations in time, and further

proposed a novel High Dimensional Harmonic Balance (HDHB) method for analyzing the limit cycle aeroelastic vibration of

airfoil [5] . Dai et al. proposed a spectral collocation method named the Time Domain Collocation (TDC) method [6] using

harmonics as trial functions, and Dirac delta functions as test functions, in the time domain. Since they assume the trial

solutions to be Fourier series, the HB, HDHB, and TDC methods are all designed to only capture periodic responses. Using

Radial Basis Functions (RBF) as trial approximations in a finite time interval, Elgohary et al. proposed the RBF collocation

method [7] and applied it to solve nonlinear initial value and two-point boundary value problems directly. All the above-

mentioned methods lead to nonlinear algebraic equations (NAEs) to be further solved. Finite element based approaches in

time domain are applied to nonlinear dynamical systems by Betsch and Steinmann [8] . A unified theory of direct time-

discretization computational algorithms, from the point of view of a generalized time weighted residual solution of the

semi-discrete equations in time, is introduced by Tamma et al. [9] . Using this unified theory, they reinterpreted the direct

time integration methods [10] , the mode superposition type methods [11] , the space and time finite element methods [12] ,

and hybrid methods [13] as special cases. The finite difference methods involve explicit and implicit type of methods. The

most common and fundamental methods using finite differences are the central difference method [14] , the average and

linear acceleration method of Newmark [15] , and the Runge–Kutta method [16,17] . An extension of Newmark- β method

is made by Hilber et al, which is currently popular as the Hilber–Hughes–Taylor (HHT)- α method [18] . In 1950, Houbolt

et al. proposed the now famous Houbolt method [19] with a multi-step formulation. Due to the lack of a well-defined

starting procedure of the original Houbolt method, Soroushian et al. proposed a one-step conversion of the original method

and provided a unified starting procedure in [20,21] . The finite difference methods are easy to implement, but often lack

stability in computations using large size steps. The implementation of an implicit method results in nonlinear algebraic

equations of unknown states, which are usually solved with Newton like methods. For some typical implicit methods, such

as the Newmark- β methods [15] and the Hilber–Hughes–Taylor- α method [18] , the stability is guaranteed only for linear

problems. For nonlinear problems, the stability is doubtful and often depends on the specific problem and the step size

adopted in the method. With relatively large size steps, some implicit methods could become so unstable and inaccurate

that they are obviously unsuited for numerical integration to predict the long-term aperiodic nonlinear dynamical behaviors

[22] . In literature, there exist some renowned energy preserving methods, such as the above-mentioned time finite element

method proposed by Betsch and Steinmann [8] , and the symplectic schemes proposed by Simo et al. [23] . Unfortunately,

these methods are restricted to Hamiltonian systems, and as far as the authors know, they cannot be extended to general

nonlinear dynamical problems such as the harmonically forced and damped buckled beam problem investigated in this

paper, with emphasis on bifurcations and chaos.

The present paper concerns mainly with the phenomena of bifurcation and chaos in structural dynamics, governed by a

semi-discrete nonlinear system of equations in time, after a spatial discretization has been carried out. Analyses of nonlinear

structural dynamics problems are still widely carried out in current engineering practice, by using the direct time-stepping

integration methods such as the Newmark- β and its modification with added damping, the Hilber-Hughes-Taylor- α methods,

wherein Taylor series expansions of the displacement and velocity vectors are used over a very small time step �t , around

the generic time t i . On the other hand, the three Optimal-Feedback-Accelerated Picard Iteration (OFAPI) algorithms introduced

in this paper, involve finitely large time intervals ( t i +1 − t i ) which are several hundreds of times larger than the time step �t

used in finite difference methods such as the Newmark and the Hilber-Hughes-Taylor- αmethods. The presently introduced

OFAPI methods represent a generalization and further improvement of the variational iteration methods developed by the

authors in [24,25] and which were initially applied to orbital mechanics problems. Furthermore, the present OFAPI methods

are based on optimal feedback Picard iteration in conjunction with the collocation concepts, instead of simply using collo-

cation in time domain directly on the nonlinear semi-discrete equations in time and using various trial functions. As will be

shown later in this paper, the OFAPI methods not only generalize the conventional collocation method and provide a new

class of methods, but also can bypass the often-difficult problem of solving the NAEs.

In Section 2 , the OFAPI algorithms for nonlinear structural dynamics are presented in easily implementable forms,

wherein the solution vector consisting of structural displacements ( x ) and structural velocities ( x ′ ) at any time t , in a large

time interval ( t i +1 − t i ) is corrected by an optimally weighted (with the weighting matrix λ) integral of the error in the solu-

tion vector from t i to t . In Section 3 , which concerns the energy preserving properties of the proposed method, the unforced

and undamped Duffing oscillator is taken as an example, where the computational error of the Hamiltonian is recorded. It

is shown that the Optimal–Feedback–Accelerated Picard Iteration algorithms achieve the best performance in preserving the

Hamiltonian, when compared with the HHT- α method and the ode45 function in MATLAB. Much more detailed numerical

investigations of OFAPI methods are made in Sections 4 and 5 with regards to the nonlinear vibration of a buckled beam.

The nonlinear vibration of a buckled beam is very representative in nonlinear structural dynamics and has attracted a

number of researchers [26–28] . Usually the spatial dynamics of a buckled beam can be well modelled by finite element,

56 X. Wang et al. / Commun Nonlinear Sci Numer Simulat 65 (2018) 54–69

boundary element, or meshless methods. However, it is shown in [29] that the clamped-clamped buckled beam model

can also be well discretized spatially by a four-mode approximation using the spatial Galerkin method. In this paper, the

Optimal–Feedback–Accelerated Picard Iteration algorithms are used to investigate the nonlinear vibrations of a buckled beam

which may exhibit bifurcation, jump phenomena, and chaos. The validity of the four-mode buckled beam model is first ex-

amined, and the nonlinear dynamical behaviors such as period doubling process and chaotic motion are then investigated.

The numerical results obtained by Optimal–Feedback–Accelerated Picard Iteration algorithms are compared with those ob-

tained by finite difference methods. It is shown that the proposed algorithms require several orders of magnitude of less

computational time than the currently widely used explicit as well as implicit methods such as the Newmark and the

Hilber–Hughes–Taylor algorithms, and achieve better accuracy and convergence speed, in problems involving bifurcation

and chaos in nonlinear structural dynamics.

2. An elucidation of the Optimal-Feedback-Accelerated Picard Iteration algorithms

In nonlinear structural dynamics modeling, the vibration of a structure is often described by a system of second-order

nonlinear ordinary differential equations after spatial discretization has already been carried out, where x is the vector of

generalized displacements, and x ′′ are the accelerations, M is the mass matrix, C is the damping matrix, N is the vector of

nonlinear restoring forces which may depend on x and the velocity vector x ′ , and F is the external force.

Mx

′′ + Cx

′ + N ( x

′ , x ) = F (t) . (1)

Eq. (1) can be further rewritten as a system of nonlinear first order ODEs: [x

′ 1

x

′ 2

]=

[x 2

−M

−1 C x 2 − M

−1 N ( x 1 , x 2 ) + M

−1 F (t)

], (2)

where N denotes the nonlinear terms, x 1 and x 2 represent displacement and velocity vector of the motion. Eq. (2) may be

considered as the “mixed-form” equivalent of Eq. (1) , wherein x 1 is the displacement vector and x 2 is the velocity vector.

2.1. Optimal error-Feedback Accelerated Picard Iteration (OFAPI) concepts

Let u 1 and u 2 be the trial functions for x 1 and x 2 in a finitely large local time interval t ∈ [ t i , t i +1 ] . By substituting them

into Eq. (2) , the error residual function is obtained as

R (t) =

[u

′ 1

u

′ 2

]+

[−u 2

M

−1 C u 2 + M

−1 N ( u 1 , u 2 ) − M

−1 F (t)

], t ∈ [ t i , t i +1 ] . (3)

To optimally correct the approximate solution at t , a simple mechanism of an optimally weighted feedback of the error

is adopted herein, which has the expression: [u 1 (t) u 2 (t)

]c

=

[u 1 (t) u 2 (t)

]+

∫ t

t i

λ(τ ) { R (τ ) } dτ , t ∈ [ t i , t i +1 ] , (4)

where the subscript c on the left-hand side indicates the “corrected solution”. In Eq. (4) , λ( τ ), a matrix, is the set of optimal

weighting functions for the feedback of the error residual R ( τ ). Eq. (4) indicates that the solution vector [ u 1 ( t ); u 2 ( t )] at any

time t in the interval t i ≤ t ≤ t i +1 is corrected by an optimally weighted error residual from time t i to t .

The iteration formula of the original Picard iteration method [30,31] can be regarded as a special case of Eq. (4) , where

λ( τ ) is simply selected as the negative unit matrix λ(τ ) = −I . Although the original Picard iteration method may converge,

it is not the most efficient approach, since λ(τ ) = −I is selected too roughly. The derivation of the optimal λ( τ ) is, however,

as follows.

For u 1 and u 2 in the neighborhood of the true solutions, i.e. u 1 = u

true 1

+ δu 1 , u 2 = u

true 2

+ δu 2 , λ is expected to be opti-

mally determined so that [u 1 (t) u 2 (t)

]c

=

[u

true 1 (t)

u

true 2 (t)

]. (5)

It means that the variation of Eq. (4) should equals to be zero if u 1 = u

true 1

, u 2 = u

true 2

, which leads to:

δ

[u 1

u 2

]c

= δ

[u 1

u 2

]+

∫ t

t i

λδ{ R (τ ) } dτ +

∫ t

t i

δλ{ R (τ ) } = 0 , (6)

Since u 1 and u 2 in Eq. (6) are supposed to be the true solutions, we have

R (τ ) =

[u

′ 1

u

′ 2

]+

[−u 2

M

−1 C u 2 + M

−1 N − M

−1 F (τ )

]= 0 . (7)

X. Wang et al. / Commun Nonlinear Sci Numer Simulat 65 (2018) 54–69 57

Therefore, Eq. (6) leads to:

δ

[u 1

u 2

]c

= δ

[u 1

u 2

]+

∫ t

t i

λδ

{[u

′ 1

u

′ 2

]+

[−u 2

M

−1 C u 2 + M

−1 N − M

−1 F (t)

]}dτ

= (I + λ) δ

[u 1

u 2

]+

∫ t

t i

{−λ′ + λJ

}δ

[u 1

u 2

]dτ = 0 , (8)

where

J =

[0 −I

M

−1 ( ∂N / ∂ u 1 ) M

−1 C + M

−1 ( ∂N / ∂ u 2 )

]. (9)

In Eq. (8) , there are variations both inside and outside the integral over time, thus the variations are made to be zero

separately by enforcing the weighting function matrix λ to satisfy the following Euler-Lagrange conditions:

λ(t) + I = 0 , and − λ′ (τ, t) + λ(τ, t) J (τ ) = 0 , τ ∈ [ t i , t] (10)

Obviously, λ is related to u 1 , u 2 , which are supposed to be u

true 1

and u

true 2

. However, the true solutions are not known in

advance. Considering that, we calculate λ in the present paper using approximated solutions instead of the true solutions

in the implementation of the algorithms, which are labeled as OFAPI algorithms for convenience.

2.2. Large time interval [ t i , t i +1 ] orthogonal polynomial collocation

Further, the following weak formulation of Eq. (4) can be established, using a matrix of test functions v ( t ).

∫ t i +1

t i

v (t)

[u 1 (t) u 2 (t)

]c

dt =

∫ t i +1

t i

v (t)

{[u 1 (t) u 2 (t)

]+

∫ t

t i

λ{ R (τ ) } dτ

}dt . (11)

In this formula, v ( t ) are test functions, the same as those in the classical weighted residual methods. Let v ( t ) be a diagonal

matrix v = diag[ v , v , ... ] , and v be Dirac Delta function for a group of collocation points t m

i , in the finitely large time interval

t i to t i +1 . For simplicity, the collocation points t m

i are simply denoted as t m

in the following parts.

v = δ(t − t m

) , t m

∈ [ t i , t i +1 ] , m = 1 , 2 , ..., M, (12)

then Eq. (11) becomes [u 1 ( t m

) u 2 ( t m

)

]c

=

[u 1 ( t m

) u 2 ( t m

)

]+

∫ t m

t i

λ{ R (τ ) } dτ , t m

∈ [ t i , t i +1 ] , m = 1 , ..., M. (13)

Eq. (13) may be interpreted as the correction of the error at each collocation point t m

, with the error residual being optimally

weighted by λ.

By using a set of orthogonal polynomials � = { φ0 , φ1 , φ2 , ... } T as basis functions, the trial functions u 1 and u 2 can be

constructed as

u 1 ,e =

N ∑

n =0

a 1 ,e,n φn , and u 2 ,e =

N ∑

n =0

a 2 ,e,n φn , (14)

Where u 1, e and u 2, e are elements of u 1 and u 2 respectively. a 1, e, n and a 2, e, n ar e coefficients t o be determined. From

Eq. (14) , we have

[ u p,e ( t 1 ) , u p,e ( t 2 ) , ..., u p,e ( t M

)] T = B [ a p,e, 0 , a p,e, 1 , ..., a p,e,N ] T , p = 1 , 2 , (15)

and

[ u p,e ′ ( t 1 ) , u p,e

′ ( t 2 ) , ..., u p,e ′ ( t M

)] T = LB [ a p,e, 0 , a p,e, 1 , ..., a p,e,N ] T , p = 1 , 2 , (16)

where

B =

⎡

⎢ ⎢ ⎣

φ0 ( t 1 ) φ1 ( t 1 ) · · · φN ( t 1 ) φ0 ( t 2 ) φ1 ( t 2 ) · · · φN ( t 2 )

. . . . . .

. . . . . .

φ0 ( t M

) φ1 ( t M

) · · · φN ( t M

)

⎤

⎥ ⎥ ⎦

(N+1) ×M

, LB =

⎡

⎢ ⎢ ⎣

φ0 ′ ( t 1 ) φ1

′ ( t 1 ) · · · φN

′ ( t 1 )

φ0 ′ ( t 2 ) φ1

′ ( t 2 ) · · · φN

′ ( t 2 )

. . . . . .

. . . . . .

φ0 ′ ( t M

) φ1 ′ ( t M

) · · · φN ′ ( t M

)

⎤

⎥ ⎥ ⎦

(N+1) ×M

(17)

Normally, the number of collocation points M is selected as the same as the number of basis functions N + 1 . It can be

found that the values of u p, e ′ at collocation points are related with those of u p, e by

[ u p,e ′ ( t 1 ) , u p,e

′ ( t 2 ) , ..., u p,e ′ ( t M

)] T = ( LB ) B

−1 [ u p,e ( t 1 ) , u p,e ( t 2 ) , ..., u p,e ( t M

)] T . (18)

58 X. Wang et al. / Commun Nonlinear Sci Numer Simulat 65 (2018) 54–69

Fig. 1. Schematic presentation of large time collocation in the interval t i ≤ t ≤ t i +1 .

In an analogous way, the values of ∫ t

t i u p,e dτ at collocation points t m

, m = 1 , 2 , ..., M can also be obtained from those of

u p, e through the following transformation. [∫ t 1

t i

u p,e dτ ,

∫ t 2

t i

u p,e dτ , ...,

∫ t M

t i

u p,e dτ

]T

= ( L −1 B ) B

−1 [ u p,e ( t 1 ) , u p,e ( t 2 ) , ..., u p,e ( t M

)] T , (19)

where

L −1 B =

⎡

⎢ ⎢ ⎢ ⎣

∫ t 1 t i

φ0 dτ∫ t 1

t i φ1 dτ · · · ∫ t 1

t i φN dτ∫ t 2

t i φ0 dτ

∫ t 2 t i

φ1 dτ · · · ∫ t 2 t i

φN dτ. . .

. . . . . .

. . . ∫ t M t i

φ0 dτ∫ t M

t i φ2 dτ · · · ∫ t M

t i φN dτ

⎤

⎥ ⎥ ⎥ ⎦

(N+1) ×M

. (20)

A schematic representation of the large time collocation method is given in Fig. 1 , wherein the time interval ( t i +1 − t i )

can be (as later shown) several hundreds of times larger than the step size �t of say, the HHT- α algorithm, for instance.

In each time interval t i to t i +1 , M collocation points are selected to interpolate the approximated solution. In this paper, the

two ends of the time interval are selected as the first and the last collocation points, i.e. t 1 i

= t i , t M

i = t i +1 . It should be noted

that even the time step between two neighboring collocation points can be larger than the time step �t used in the HHT- αalgorithm, for instance.

2.3. Optimal-Feedback-Accelerated Picard Iteration (OFAPI) algorithms

Although we have already obtained the iterative formula (13) through collocation, the matrix of generalized Lagrange

multipliers λ in it remains a puzzle. Directly solving the constraint Eq. (10) for λ is unlikely for most nonlinear cases.

The following introduces three approaches to bypass this dilemma, from which 3 OFAPI algorithms are developed. The first

algorithm OFAPI-1 is mathematically equivalent to Eq. (13) , but it involves inversion of Jacobian matrix. OFAPI-2 and 3

approximate λ with truncated polynomial and exponential series respectively, thus no matrix inversion is required therein.

Although the OFAPI algorithms-1, 2, and 3 are derived from the same optimal error-feedback iteration concept, they have

some differences in implementation and computational performances that the users of them should be aware of. As will

be shown in the flow chart overview of these three algorithms in Fig. 2 , in the implementations of OFAPI algorithms-1

and 3, the initial conditions have to be incorporated in the iteration formula and the excess collocation equations need

to be removed. In OFAPI algorithm-2, the initial conditions are naturally satisfied, thus the implementation of it is more

straightforward than OFAPI algorithms-1 and 3.

The convergence speed of the OFAPI algorithm-1 is supposed to be the fastest, since it is equivalent to Eq. (12) , where

the matrix of weighting functions λ is optimally derived. However, the OFAPI algorithm-1 involves inversion of the Jacobian

matrix ( E +

J ) , which is varying during the iteration process. For systems with high dimensions, computing the inversion of

X. Wang et al. / Commun Nonlinear Sci Numer Simulat 65 (2018) 54–69 59

Fig. 2. Flow chart overview of the OFAPI algorithms.

the Jacobian matrix could be very time consuming. In addition, if the Jacobian matrix becomes ill-conditioned during the

iteration, the OFAPI algorithm-1 may easily diverge. On the contrary, the OFAPI algorithms-2 and 3 are free from inverting

matrices during the iteration process.

The formulations of these 3 algorithms are presented in this section. Further detailed comparisons between them are

made in Section 3 and 4 through numerical simulations.

60 X. Wang et al. / Commun Nonlinear Sci Numer Simulat 65 (2018) 54–69

2.3.1. OFAPI algorithm-1

Differentiating Eq. (4) leads to

d

dt

[u 1 (t) u 2 (t)

]c

=

d

dt

[u 1 (t) u 2 (t)

]+ λ(t) R (t) +

∫ t

t i

∂λ

∂t R (τ ) dτ , (21)

where R ( τ ) is the residual function in Eq. (3) . According to Eq. (10) and using the theory of Magnus series, it is proved

[17] that

λ(t) = −I , and

∂λ

∂t = −J (t) λ. (22)

Substituting them into Eq. (21) , we have

d

dt

[u 1 (t) u 2 (t)

]c

= −[−u 2

M

−1 C u 2 + M

−1 N − M

−1 F

]− J (t)

∫ t

t i

λ

{[u

′ 1

u

′ 2

]+

[−u 2

M

−1 C u 2 + M

−1 N − M

−1 F

]}dτ . (23)

Noting that ∫ t

t i λ{[u

′ 1

u

′ 2

]+

[−u 2

M

−1 C u 2 + M

−1 N − M

−1 F (t)

]}dτ =

[u 1

u 2

]c −

[u 1

u 2

]accor ding to the optimal error-feedback itera-

tion formula (4) , Eq. (23) is rewritten as

d

dt

[u 1 (t) u 2 (t)

]c

= −[−u 2

M

−1 C u 2 + M

−1 N − M

−1 F

]− J (t)

{[u 1

u 2

]c

−[

u 1

u 2

]}. (24)

After rearrangement, it is rewritten as

d

dt

[u 1 (t) u 2 (t)

]c

+ J (t)

[u 1 (t) u 2 (t)

]c

= −[−u 2

M

−1 C u 2 + M

−1 N − M

−1 F

]+ J ( t )

[u 1 ( t ) u 2 ( t )

]. (25)

Eq. (25) can be regarded as equivalent to Eq. (4) . By collocating in the local time interval [ t i , t i +1 ] , an algebraic iterative

formula is obtained as. [u 1

′ ( t m

) u 2

′ ( t m

)

]c

+ J ( t m

)

[u 1 ( t m

) u 2 ( t m

)

]c

= −[−u 2 ( t m

) M

−1 C u 2 ( t m

) + M

−1 N ( t m

) − M

−1 F ( t m

)

]+ J ( t m

)

[u 1 ( t m

) u 2 ( t m

)

], (26)

where m = 1 , 2 , ...M, t m

∈ [ t i , t i +1 ] , and t i +1 − t i is a finite large time interval.

After rearranging the sequence of the collocation equations and using the relationship in Eq. (18), Eq. (26) is rewritten

as

( E +

J )

[U 1

U 2

]c

= −[−U 2

˜ M

−1 ˜ C U 2 +

˜ M

−1 ˜ N − ˜ M

−1 ˜ F

]+

J

[U 1

U 2

], (27)

where

U p =

[u p, 1 ( t 1 ) u p, 1 ( t 2 ) ... u p, 1 ( t M

) u p, 2 ( t 1 ) u p, 2 ( t 2 ) ... u p, 2 ( t M

) ... ]T

, p = 1 , 2 . (28)

The configuration of the matrices ˜ E , J , ˜ M , ˜ C , ˜ N , ˜ F are provided in Appendix C .

It should be noted that the initial conditions are not incorporated in Eq. (27) . For that, without loss of generality, we usu-

ally select the first collocation point at the initial boundary, of which the values u p, e ( t 1 ) are given. By doing that, Eq. (27) be-

comes overdetermined, thus it is necessary to drop excess collocation equations at time t 1 . Finally, Eq. (27) is modified as

[U 1

U 2

]c

r

=

[U 1

U 2

]r

− ( E

r +

J r ) −1

{˜ E

[U 1

U 2

]+

[−U 2

˜ M

−1 ˜ C U 2 +

˜ M

−1 ˜ N − ˜ M

−1 ˜ F

]}r

, (29)

The symbol [ •] r denotes the remained matrix after the (lM + 1) th rows and columns in [ •] are dropped, l = 0 , 1 , 2 , ... . If

[ •] r is a vector, we just need to remove the (lM + 1) th rows in [ •].

A flow diagram is presented in Fig. 2 to illustrate the OFAPI algorithm-1 along with the other two algorithms.

2.3.2. OFAPI algorithm-2

According to Eq. (10) , the Lagrange multipliers can be approximated by Taylor series as

λ(τ ) = T 0 + T 1 (τ − t) + T 2 (τ − t) 2 + ..., (30)

where T 0 = −I , T 1 = −J (t) , T 2 = −J (t) 2 / 2 , and so on. To generally obtain the Taylor series approximation of λ( τ ), one may

use the Differential Transform Method (DTM) [32] . Substituting it into Eq. (13) , the correctional formula is obtained as

X. Wang et al. / Commun Nonlinear Sci Numer Simulat 65 (2018) 54–69 61

[u 1 ( t m

) u 2 ( t m

)

]c

=

[u 1 ( t m

) u 2 ( t m

)

]+

∫ t m

t i

[−I − J (t)(τ − t) − J (t)

2

2

(τ − t) 2 + · · ·

]{[u

′ 1

u

′ 2

]+

[−u 2

M

−1 C u 2 + M

−1 N − M

−1 F

]}dτ

(31)

In implementations, λ( τ ) is commonly approximated by truncated Taylor series [24] . The simplest could be the zeroth

order approximation

λ(τ ) = −I , (32)

or the first order approximation

λ(τ ) = −I − J (t)(τ − t) . (33)

Higher order approximations are possible, but they are rarely used in practice considering the computational complexity.

With the zeroth order approximation of λ( τ ), Eq. (31) becomes [u 1 ( t m

) u 2 ( t m

)

]c

=

[u 1 ( t m

) u 2 ( t m

)

]−

∫ t m

t i

{[u

′ 1

u

′ 2

]+

[−u 2

M

−1 C u 2 + M

−1 N − M

−1 F

]}dτ . (34)

With the first order approximation of λ( τ ), Eq. (31) becomes [u 1 ( t m

) u 2 ( t m

)

]c

=

[u 1 ( t m

) u 2 ( t m

)

]+

∫ t m

t i

[ −I − J ( t m

)(τ − t m

)]

{[u

′ 1

u

′ 2

]+

[−u 2

M

−1 C u 2 + M

−1 N − M

−1 F

]}dτ . (35)

where m = 1 , 2 , ...M, t m

∈ [ t i , t i +1 ] , and t i +1 − t i is a finite large time interval. By separating the terms involving t m

and τ ,

Eq. (35) leads to [u 1 ( t m

) u 2 ( t m

)

]c

=

[u 1 ( t m

) u 2 ( t m

)

]−

∫ t m

t i

R (τ ) dτ + J ( t m

) t m

∫ t m

t i

R (τ ) dτ − J ( t m

)

∫ t m

t i

τR (τ ) dτ , (36)

After some rearrangements, Eq. (36) can be rewritten as [U 1

U 2

]c

=

[U 1

U 2

]− ˜ H

R +

J T

H

R − ˜ J H

T

R , (37)

where ˜ H is the transformation matrix corresponding to integral, and

˜ T is the matrix related with time t . The configurations

of matrices ˜ H , ˜ T , J and

˜ R are provided in Appendix C . Fig. 2 shows the flow chart of OFAPI algorithm-2.

2.3.3. OFAPI algorithm-3

Considering Eq. (23) , if the Lagrange multiplier λ is approximated by Taylor series, we have

d

dt

[u 1 (t) u 2 (t)

]c

= −[−u 2

M

−1 C u 2 + M

−1 N − F

]− J (t)

∫ t

t i

[ T 0 + T 1 (τ − t) + ... ]

{[u

′ 1

u

′ 2

]+

[−u 2

M

−1 C u 2 + M

−1 N − M

−1 F

]}dτ

(38)

If λ is simply approximated by T 0 , Eq. (38) becomes

d

dt

[u 1 (t) u 2 (t)

]c

= −[−u 2

M

−1 C u 2 + M

−1 N − M

−1 F

]− J (t)

∫ t

t i

T 0

{[u

′ 1

u

′ 2

]+

[−u 2

M

−1 C u 2 + M

−1 N − M

−1 F

]}dτ. (39)

Since T 0 = −I , Eq. (39) is further rewritten as

d

dt

[u 1 (t) u 2 (t)

]c

= −[−u 2

M

−1 C u 2 + M

−1 N − M

−1 F

]+ J (t)

∫ t

t i

{[u

′ 1

u

′ 2

]+

[−u 2

M

−1 C u 2 + M

−1 N − M

−1 F

]}dτ (40)

By using u p ( p = 1 , 2 ) as trial functions and making collocations, the OFAPI algorithm-3 is obtained from Eq. (40) as [u 1

′ ( t m

) u 2

′ ( t m

)

]c

= −[−u 2 ( t m

) M

−1 C u 2 ( t m

) + M

−1 N ( t m

) − M

−1 F ( t m

)

]+ J ( t m

)

∫ t m

t i

{[u

′ 1

u

′ 2

]+

[−u 2

M

−1 C u 2 + M

−1 N − M

−1 F

]}dτ

(41)

where u p ′ ( t m

) and the integrals can be obtained as is stated in Eqs. (18 ) and ( 19 ). After rearrangements, Eq. (41) can be

written as

˜ E

[U 1

U 2

]c

= −[−U 2

˜ M

−1 ˜ C U 2 +

˜ M

−1 ˜ N − ˜ M

−1 ˜ F

]+

J H

R . (42)

The configurations of matrices ˜ E , ˜ M , ˜ C , ˜ N , J , ˜ H , and

˜ R are provided in Appendix C .

62 X. Wang et al. / Commun Nonlinear Sci Numer Simulat 65 (2018) 54–69

Fig. 3. Computational error of Hamiltonian using the HHT- α and the 3 OFAPI algorithms, and ode45 in MATLAB.

Table 1

Comparison of HHT, OFAPI and ode45 on solving Duffing equation.

Computational time (s) Iteration steps Step size

HHT, α= 0 24 299,827 0.01

HHT, α= −0 . 1 26 298,448 0.01

OFAPI 2 6076 2

ode45(MATLAB) 11 1,033,129 0.001

Note that the initial conditions are not guaranteed by Eq. (42) , just like Eq. (27) in OFAPI algorithm-1, thus it should be

further modified as

˜ E

r

[U 1

U 2

]c

r

=

{−[−U 2

˜ M

−1 ˜ C U 2 +

˜ M

−1 ˜ N − ˜ M

−1 ˜ F

]+

J H

R

}r

− ˜ E

d

[U 1

U 2

]c

d

, (43)

The symbol [ •] r denotes the remained matrix after the (lM + 1) th rows and columns in [ •] are dropped, l = 0 , 1 , 2 , ... .

If [ •] r is a vector, we just need to remove the (lM + 1) th rows in [ •]. The symbol [ •] d denotes the part of [ •] that were

dropped to obtain [ •] r . The flow chart of this algorithm is provided in Fig. 2 .

2.4. Performance of OFAPI on energy preservation

To investigate the energy conservation properties of the HHT- α and the 3 OFAPI algorithms, a simple undamped duffing

equation is used for demonstration.

x ′′ − x + x 3 = 0 .

The Hamiltonian energy of this system is

H =

x ′ 2

2

− x 2

2

+

x 4

4

.

Starting from the initial state x (0) = 1 . 5 , x ′ (0) = 0 , the system is integrated using the HHT- α, the 3 OFAPI, and the

ode45 methods. The step size of HHT- α method is �t = 0 . 01 and the simulation is carried in the time interval t ∈ [0, 10 0 0].

For the 3 OFAPI algorithms, the step size is selected as �t = 2 , with 32 collocation points in each step. The absolute and

relative accuracies of ode45 are both set as 1 E − 15 . The computational errors of the Hamiltonian are recorded and plotted

in Fig. 3 . It can be seen that both the OFAPI algorithms and the ode45 are much superior to the HHT- α algorithm on

energy conservation. Notably, the present 3 OFAPI algorithms behave even better than ode45, with a negligible error of

Hamiltonian of 1 E − 13 . Among all these methods, the HHT- α method with α = −0 . 1 is the worst on energy conservation,

as can be seen in Fig. 3 (a). After dropping out the numerical damping by setting α = 0 , the performance of HHT- α method

is much improved, with the error of Hamiltonian being 1 E − 5 .

Aside from the much higher computational accuracy (energy preservation), the OFAPI is also much more efficient as

shown in Table 1 .

X. Wang et al. / Commun Nonlinear Sci Numer Simulat 65 (2018) 54–69 63

3. Nonlinear vibrations of a buckled beam

The governing equation of a buckled one-dimensional Bernoulli beam is written in non-dimensional form as

w + w

i v + P w

′′ + c ˙ w − 1

2

w

′′ ∫ 1

0

w

′ 2 dx = F (x ) cos t, (44)

BC ’ s : w = w

′ = 0 at x = 0 and x = 1 ,

where w is the transverse displacement of the beam, and P is the axial load on the beam. F ( x ) is the transverse distributed

load on the beam and is the frequency of the applied load F ( x ). The overdot denotes the derivative with respect to time

t , while the prime denotes the derivative with respect to the spatial coordinate x . To semi-discretize Eq. (44) , the mode

decomposition method introduced by Emma et al. [29] is applied herein.

To solve the preceding partial differential equation, we first assume the spatial modes:

w (x, t) = w s (x ) + v (x, t) = w s (x ) +

N ∑

n =1

φn (x ) q n (t) , (45)

where w s is the static postbuckling displacement, v ( x, t ) is the superposed dynamic response around the buckled configura-

tion. φn ( x ) are the mode shapes of vibration around the buckled configuration, and q n ( t ) are the amplitudes of φn ( x ).

The buckled configuration w s ( x ) can be obtained by first solving the static buckling problem where the time derivatives

and the dynamic load are removed in Eq. (44) .

w

i v + P w

′′ − 1

2

w

′′ ∫ 1

0

w

′ 2 dx = 0 . (46)

Mathematically, there could be various buckled mode shapes, depending on the corresponding load. However, in struc-

tural mechanics, the first buckled mode shape is mostly of importance, from which w s ( x ) is obtained as [29]

w s (x ) =

1

2

b(1 − cos 2 πx ) . (47)

The non-dimensional transverse displacement b at the mid-span of the beam is related to the load P via [29]

b 2 = 4(P − P c ) / π2 , (48)

where P c is the critical load corresponding to the first Euler buckled mode, namely P c = 4 π2 .

Substituting the assumed solution of w ( x, t ) into the governing equation and dropping all the nonlinear, damping, and

forcing terms, we have the following linear partial differential equation that can be tackled using the linear vibration mode

theory.

v + v i v + 4 π2 v ′′ − 2 b 2 π3 cos 2 πx

∫ 1

0

v ′ sin 2 πx dx = 0 . (49)

By assuming v (x, t) = φ(x ) e iωt and substituting it into Eq. (49) , the mode shape is obtained as

φ(x ) = φh + φp = ( c 1 sin s 1 x + c 2 cos s 1 x + c 3 sinh s 2 x + c 4 cosh s 2 x ) + c 5 cos 2 πx, (50)

where s 1 , 2 = (±2 π2 +

√

4 π2 + ω

2 ) 1 / 2 , and c 5 should satisfy the following equation:

(2 b 2 π4 − ω

2 ) c 5 = 2 b 2 π3

∫ 1

0

φ′ h sin 2 πx dx. (51)

Using the boundary conditions and Eq. (51) , the mode shapes φn ( x ) can be obtained.

The resulting linear vibration mode shapes φn ( x ) are used to construct the solution v ( x, t ). Using the multi-mode Galerkin

discretization, where the weighting functions are the same as the trial functions φn ( x ), the partial differential Eq. (44) is then

reduced to a system of coupled Duffing equations.

q n + ω

2 n q n = −c q n + b

N ∑

i, j

A ni j q i q j +

N ∑

i, j,k

B ni jk q i q j q k + f n cos t, n = 1 , 2 , ..., N. (52)

In this paper, four modes are retained in the reduced model. The buckling level is selected as b = 4 . Correspondingly

the natural frequencies of the four linear vibration modes are obtained as ω 1 = 30 . 7067 , ω 2 = 44 . 3627 , ω 3 = 108 . 3322 , and

ω 4 = 182 . 1178 . The parameters A nij , and B nijk are provided in Appendices A and B .

64 X. Wang et al. / Commun Nonlinear Sci Numer Simulat 65 (2018) 54–69

Fig. 4. Bifurcation diagrams obtained by (a) force-sweeping, (b) frequency-sweeping process. Same results obtained for HHT- α (with very small step size)

and the 3 OFAPI algorithms.

4. Numerical results and discussion

Considering the discretized nonlinear system (52) , several types of nonlinear resonances of the buckled beam may occur

due to the external harmonic excitation. The primary resonance is mostly observed when the excitation frequency is

close to one of the mode frequencies ω n , which often leads to a periodic motion of large amplitude in that mode. For the

existence of quadratic nonlinearities and cubic nonlinearities, the subharmonic resonances and superharmonic resonances

may also occur for that has an integer relationship to ω n .

In the following, the resonance of the buckled beam under harmonic excitations is investigated. The frequency is

selected as being close to the natural frequency of the first vibration mode ω 1 . The external force is supposed to be uniform

over the length of the beam, thus F ( x ) in Eq. (44) is constant. Through Galerkin discretization, the forces imposed on the

four linear vibration modes are obtained as f 1 = −0 . 850654 F , f 2 = 0 , f 3 = 0 . 309884 F , and f 4 = 0 .

In the numerical simulations, the force-sweep as well as the frequency-sweep processes are used to obtain an overview

of the nonlinear dynamical behaviors of the buckled beam when subjected to a primary resonance excitation of its first

vibration mode. Considering that the magnitudes of the coefficients ω

2 n , bA nij , and B nijk are roughly between 10 3 and 10 4

in the restoring forces, the amplitude of the external force F is set to vary between F = 40 and F = 600 in the force-sweep

process . In the frequency-sweep process, F is fixed at F = 400 , while the frequency is varied between = 30 and = 28 .

The numerical methods including HHT- α (with the values of α selected as 0 and −0.1), OFAPI, and ode45 function built

in MATLAB are used to investigate the buckled beam system. In this paper, the basis functions used in OFAPI method are

selected as the Chebyshev polynomials of the first kind and the collocation points are selected as the Chebyshev-Gauss-

Lobatto nodes.

4.1. Nonlinear dynamical behaviors including bifurcation and chaos

With the discretized equations derived above, the nonlinear vibrations of a buckled beam are first investigated under

uniform harmonic excitations, in which the frequency of the external load F ( x )cos t is = 30 . A force sweeping approach

is first adopted herein to capture the bifurcation phenomenon. For simplicity, a periodic motion is referred to as period- n

motion if its period is nT , where T = 2 π/ . It is shown in Fig. 4 (a) that a period-one motion is obtained for F = 40 . As

the excitation force F increases, the first period-doubling bifurcation occurs at about F = 420 . By further increasing the

force amplitude, the second period-doubling occurs at about F = 454 . The next bifurcation occurs at F = 461 , leading to the

period-eight motion. Similar to the force sweeping approach, a period doubling bifurcation route to chaos is also revealed

by using frequency sweeping approach in Fig. 4 (b) with the amplitude fixed at F = 400 .

The results in Fig. 4 can be obtained by using both the HHT- α and the 3 OFAPI algorithms. However, the time step

size has to be selected very small to accurately obtain the steady periodic motions in using the HHT- α algorithm, of which

the computational time (as shown in detail in Table 3 to follow) will be much more prolonged. On the contrary, the step

size of the present OFAPI algorithms can be selected to be relatively very large (several hundreds of times larger than the

time step in the HHT- α, as shown in Table 3 to follow), and all the 3 OFAPI can easily achieve high accuracy in predicting

these limit cycle oscillations. It will be shown in the next subsection that all the 3 OFAPI algorithms have a far better

performance than the HHT- α algorithm in terms of accuracy, computational time, and speed of convergence, in predicting

the nonlinear dynamical responses of the buckled beam involving bifurcation and chaos.

X. Wang et al. / Commun Nonlinear Sci Numer Simulat 65 (2018) 54–69 65

Fig. 5. The same chaotic motion revealed by HHT- α (with very small step size) and the 3 OFAPI algorithms. (a) Phase portrait, (b) time responses,

(c) Poincare map, (d) FFT curves.

As the period-doubling bifurcation proceeds, more sinks appear in the chaotic regime of Poincare map of the dynamical

system, and eventually the motion becomes completely chaotic. In Fig. 5 , the chaotic motion is presented through phase

portrait, response curve, Poincare map, and FFT curve. Both the HHT- α method with damping coefficients α = −0 . 1 , and the

OFAPI algorithms are used to capture the chaotic motion. The step size of HHT method is �t = T / 10 0 0 , while that of OFAPI

method is T /5, where T is the period of the first vibration mode, T = 2 π/ ω 1 .

4.2. Comparison between the HHT- α and the 3 OFAPI algorithms

The discretized model is solved using both the HHT- α and the 3 OFAPI algorithms. It is found that all the algorithms can

predict the limit cycle oscillations and chaos. However, the computational performances of these algorithms are very differ-

ent. In the analysis below, various step sizes are used to test the stability of the algorithms. It is found through simulations

that all the 3 OFAPI algorithms converge for both periodic and chaotic motions with step sizes as large as T /5. The largest

step size of HHT- α method depends on the nonlinear algebraic equations (NAEs) solver. By using Newton-Raphson method,

it is found that the step size should be no greater than T /20, otherwise the solution of nonlinear equilibrium equations in

the HHT- α may not be found [19] .

Additional simulations were conducted to reveal the steady periodic motions of the buckled beam under different ex-

ternal excitations, using HHT- α and the 3 OFAPI algorithms separately. After the motion settles down, the extremes of the

displacement q 1 ( t ) obtained by the HHT, and the 3 OFAPI algorithms are recorded in Table 2 . The exact values are assumed

to be provided by ode45, and they are fully consistent with the results obtained by the 3 OFAPI algorithms, with the time

66 X. Wang et al. / Commun Nonlinear Sci Numer Simulat 65 (2018) 54–69

Table 2

Extremes of the steady periodic motions obtained by HHT- α and OFAPI methods.

Methods Period-1 Period-2 Period-4 Period-8

OFAPI ( T /5) 0.6006 3.0983 2.9881 2.9752

HHT- α α = −0 . 1 α = 0 α = −0 . 1 α = 0 α = −0 . 1 α = 0 α = −0 . 1 α = 0

T /5 \ \ \ \ \ \ \ \ T /10 \ \ \ \ \ \ \ \ T /50 0.5626 0.5954 3.1282 3.1164 3.0015 2.9955 \ \ T /100 0.5826 0.5995 3.1074 3.1025 2.9912 2.9898 \ 2.9828

T /500 0.5972 0.6006 3.0994 3.0984 2.9884 2.9881 2.9770 2.9755

T /10 0 0 0.5989 0.6006 3.0988 3.0983 2.9882 2.9881 2.9760 2.9756

Fig. 6. Comparisons between the limit cycle oscillations obtained by the OFAPI ( �t = T / 5 ) and HHT- α ( �t = T / 50 , α = −0 . 1 ) algorithms. (a) period-4

motion, (b) period-8 motion.

step size being T /5. The number of collocation points used in the 3 OFAPI algorithms is N = 7 . Increasing the collocation

points can further improve the accuracy of the OFAPI algorithms.

The HHT- α method fails to work for �t = T / 5 and T /10, because the Newton-Raphson iteration scheme cannot converge

for such large steps. With the step size being T /50, the HHT- α method provides steady periodic motions, although they are

quite erroneous compared to the exact ones. Particularly, in the parameter region where the period-8 motion dominates, the

HHT- α method only provides period-4 motion, which is due to the integrating inaccuracy. By shortening the step size, the

accuracy is significantly improved. However, there are still some observable discrepancies between the results of the HHT-

α and the exact ones, even with the step size in HHT being as small as �t = T / 10 0 0 . In the simulations, different values

of α are used. In the case of α = −0 . 1 , a moderate numerical damping is included, which could filter the high-frequency

component of motion and avoid divergence. For α = 0 , no numerical damping is introduced and the HHT- α method becomes

the average acceleration method. By comparing the values of extremes with the exact ones, Table 2 indicates that the

periodic motions are better predicted with α = 0 than with α = −0 . 1 . This result is reasonable since the numerical damping

introduced in HHT- α method makes the system behaves like that extra damping exists in the simulation.

The limit cycles of period-4 and period-8 motions are obtained using the 3 OFAPI algorithms with �t = T / 5 . They are

compared with the results obtained from HHT- α method, where α = −0 . 1 and �t = T / 50 . It is shown in Fig. 6 that the

HHT- α method cannot predict the true dynamical behavior with such a large step size. In Fig. 6 (a), an obvious discrepancy

exists between the results of the 3 OFAPI and the HHT- αalgorithms, while in Fig. 6 (b), the HHT- α method gives a period-

4 motion at where the period-8 motion should be revealed. In all, both the OFAPI and the HHT- α algorithms can stably

integrate the nonlinear system of buckled beam. However, the step size of HHT- α method should be selected to be very

small to obtain valid solutions, several hundreds of times smaller than in the 3 present OFAPI algorithms.

Generally, the evaluation of computational accuracy is nontrivial for numerical methods, especially in nonlinear problems.

Herein, the numerical solution of MATLAB built in ode45 function is used as the benchmark to estimate the computational

error of HHT- α and OFAPI algorithms. The absolute and the relative accuracies of ode45 are both set as 1 E − 15 . The transient

responses of period-one, period-two, period-four and period-eight motions are obtained using the HHT- α and the 3 OFAPI

algorithms respectively. Then by comparing with those obtained by ode45, the discrepancies of the numerical results are

shown in Fig. 7 . To evaluate the highest accuracy that the HHT and the 3 OFAPI algorithms may achieve, the steps size of

X. Wang et al. / Commun Nonlinear Sci Numer Simulat 65 (2018) 54–69 67

Fig. 7. The computational error of the HHT- α and the 3 OFAPI algorithms.

Table 3

Computational performance of HHT- α, the 3 OFAPI, and ode45.

Cases Computational time (s) Iteration steps Step size

HHT- α OFAPI ode45 HHT- α OFAPI ode45 HHT- α OFAPI ode45

Period-1 11.4 2.6 17.6 593,468 5392 903,797 0.0 0 01 0.035 1.6e-05

Period-2 11.8 2.5 22.5 599,664 5509 1,022,317 0.0 0 01 0.035 1.5e-5

Period-4 11.6 2.6 21.2 599,670 5608 1,015,945 0.0 0 01 0.035 1.5e-5

Period-8 11.7 2.4 19.6 599,629 5567 1,015,821 0.0 0 01 0.035 1.5e-5

HHT- α method is selected as �t = 1 e − 4 , while that of OFAPI method is �t = T / 6 = 0 . 035 with 13 collocation points in

each time step.

As is illustrated in Section 4.1 , the transient response of period-one motion is relatively simple and non-chaotic. It is

reflected by the consistent numerical discrepancies over the simulation time in Fig. 7 (a). As is shown, all the 3 OFAPI algo-

rithms achieve very high accuracy with respect to ode45. The discrepancies between them is five magnitudes lower than

that between HHT- α and ode45. For the multiple-period motions, the numerical discrepancies accumulate exponentially for

both HHT- α and OFAPI algorithms before t = 10 , as shown in Fig. 7 (b). This can be explained by the fact that chaotic regime

exists around the periodic motion. It is shown that the accuracy of all the 3 OFAPI algorithms is still much higher than the

HHT- α method in the integration of transient chaotic motions. Although the results of the OFAPI algorithms diverge from

that of ode45 after t = 10 , it does not mean that the OFAPI algorithms fail. Actually, the chaotic regime is so sensitive to the

initial state that even an error which occurs in the machine precision could blow up and cause significant discrepancy in

the final state. Through numerical simulations it is found that smaller time interval can achieve the same high accuracy as

indicated in Fig. 7 (b) with fewer collocation points and save some computational time. It is also found that the largest time

interval of OFAPI method is no more than T /4 to ensure accurate long term integration of the system (52) .

Table 3 lists the computational time, iteration steps and step sizes of the HHT- α and the OFAPI algorithms. It is found

that the OFAPI algorithms consume roughly only 20% of the computational time required by the HHT- α method. These cost

savings are expected to be much more dramatic for large scale high order dynamical systems. The time-step size used in

HHT- α is 0.0 0 01, while that in OFAPI is 0.035. According to the performances demonstrated in Fig. 7 and Table 3 , the OFAPI

algorithms are much more accurate and efficient than the implicit HHT- α method, in addition to requiring only very much

larger step sizes. For further comparison, the performance indexes of ode45 are also presented in Table 3 . It is shown that

the computational efficiency of all the 3 OFAPI algorithms is far superior to even that of ode45. Noting that the ode45 is

a built-in function in MATLAB that has already been optimized, while the OFAPI algorithms are implemented in MATLAB

with a roughly designed program. Moreover, the present OAPI algorithms can be easily improved, using parallel programing,

which will further improve the computational efficiency.

4.3. Discussions of the relative performances of the 3 OFAPI algorithms presented in this paper

Considering the buckled beam problem in this paper, the performances of the OFAPI algorithms are compared with each

other. The comparison results listed in Table 4 are obtained during the simulation time of 200 T , where T is the period of

the first mode.

68 X. Wang et al. / Commun Nonlinear Sci Numer Simulat 65 (2018) 54–69

Table 4

Comparison of OFAPI algorithms.

Method Largest step size Number of iterative steps Computational time

OFAPI algorithm-1 T /5 18,202 9s

OFAPI algorithm-2 T /5 14,765 6s

OFAPI algorithm-3 T /5 16,636 10s

As is shown in Table 4 , the largest step sizes of all the 3 stable OFAPI algorithms are the same, while the number

of iterative steps and computational time of OFAPI algorithm-2 are the least among the three proposed algorithms. The

computational errors are not provided herein because the 3 OFAPI algorithms behave similarly in terms of accuracy, which

is already presented in Fig. 7 s. Overall, although all the proposed 3 OFAPI algorithms can be efficiently applied to solving the

nonlinear dynamical systems, the OFAPI algorithm-2 is the most recommended, not only because it incorporates the initial

boundary conditions inherently and is free from inverting the Jacobian matrix, but also for that it has the best computational

performance among the proposed method.

5. Conclusion

The 3 OFAPI algorithms are used to solve a nonlinear dynamical model of the clamped-clamped buckled beam. The

numerical results show that the proposed algorithms are very accurate and efficient in predicting nonlinear dynamical be-

haviors including bifurcation and chaos. Through the force-sweep and frequency-sweep processes, the periodic motions and

the period-doubling routes to chaos are successfully captured by the OFAPI algorithms. It is also found in the simulations

that the steady multiple-periodic motions are surrounded by chaotic motions in the neighborhood. It is the reason for the

chaotic transient motions to be hardly predicted precisely.

The robustness of the OFAPI algorithms is indicated by the highly accurate and stable integration even using a large step

size �t = T / 5 . Compared with the HHT- α method, where the values of α are selected as 0 and 0.1 respectively, the proposed

OFAPI algorithms are superior on many aspects involving accuracy, efficiency, stability and energy conservation. It is shown

in this paper that the HHT- α method cannot accurately predict the periodic vibrations of the buckled beam unless extremely

small step sizes are used. The ode45 function built in MATLAB achieves relatively high accuracy, but the computational cost

is much higher than the presently proposed OFAPI algorithms. The example of unforced Duffing equation also indicates that

the energy of the system is best conserved by the proposed algorithms, compared with the HHT- α method and ode45.

Finally, the performances of the 3 OFAPI algorithms are compared with each other. It turns out that the OFAPI algorithm-2

is the most convenient and has the best computational performance among them.

Appendix A

A 1 , 1 , 1 = 124 . 905 , A 1 , 1 , 2 = 0 , A 1 , 1 , 3 = 162 . 554 , A 1 , 1 , 4 = 0 , A 1 , 2 , 2 = 171 . 098 A 1 , 2 , 3 = 0 , A 1 , 2 , 4 = −141 . 515 ,

A 1 , 3 , 3 = 407 . 072 , A 1 , 3 , 4 = 0 , A 1 , 4 , 4 = 625 . 188 ;A 2 , 1 , 1 = 0 , A 2 , 1 , 2 = 342 . 195 , A 2 , 1 , 3 = 0 , A 2 , 1 , 4 = −141 . 515 , A 2 , 2 , 2 = 0 , A 2 , 2 , 3 = 391 . 398 , A 2 , 2 , 4 = 0 ,

A 2 , 3 , 3 = 0 , A 2 , 3 , 4 = −161 . 863 , A 2 , 4 , 4 = 0 ;A 3 , 1 , 1 = 81 . 2768 , A 3 , 1 , 2 = 0 , A 3 , 1 , 3 = 814 . 145 , A 3 , 2 , 2 = 195 . 699 , A 3 , 1 , 4 = 0 , A 3 , 2 , 3 = 0 ,

A 3 , 2 , 4 = −161 . 863 , A 3 , 3 , 3 = 1264 . 72 , A 3 , 3 , 4 = 0 , A 3 , 4 , 4 = 715 . 082 ;A 4 , 1 , 1 = 0 , A 4 , 1 , 2 = −141 . 515 , A 4 , 1 , 3 = 0 , A 4 , 1 , 4 = 1250 . 38 , A 4 , 2 , 2 = 0 , A 4 , 2 , 3 = −161 . 863 , A 4 , 2 , 4 = 0 ,

A 4 , 3 , 3 = 0 , A 4 , 3 , 4 = 1430 . 16 , A 4 , 4 , 4 = 0 ;

Appendix B

B 1 , 1 , 1 , 1 = −65 . 3795 , B 1 , 1 , 1 , 3 = −79 . 2731 , B 1 , 1 , 2 , 2 = −268 . 675 , B 1 , 1 , 2 , 4 = 222 . 22 , B 1 , 1 , 3 , 3 = −600 . 138 ,

B 1 , 1 , 4 , 4 = −981 . 733 , B 1 , 2 , 2 , 3 = −108 . 59 , B 1 , 2 , 3 , 4 = 89 . 8146 , B 1 , 3 , 3 , 3 = −233 . 924 , B 1 , 3 , 4 , 4 = −396 . 786 ;B 2 , 1 , 1 , 2 = −268 . 675 , B 2 , 2 , 2 , 2 = −1104 . 11 , B 2 , 1 , 2 , 3 = −217 . 18 , B 2 , 2 , 3 , 3 = −2378 . 47 , B 2 , 1 , 1 , 4 = 111 . 11 ,

B 2 , 2 , 2 , 4 = 1369 . 81 , B 2 , 1 , 3 , 4 = 89 . 8146 , B 2 , 3 , 3 , 4 = 983 . 614 , B 2 , 2 , 4 , 4 = −4412 . 05 , B 2 , 4 , 4 , 4 = 1668 . 42 ;B 3 , 1 , 1 , 1 = −26 . 4244 , B 3 , 1 , 2 , 2 = −108 . 59 , B 3 , 1 , 1 , 3 = −600 . 138 , B 3 , 2 , 2 , 3 = −2378 . 47 , B 3 , 1 , 3 , 3 = −701 . 773 ,

B 3 , 3 , 3 , 3 = −5123 . 69 , B 3 , 1 , 2 , 4 = 89 . 8146 , B 3 , 2 , 3 , 4 = 1967 . 23 , B 3 , 1 , 4 , 4 = −396 . 786 , B 3 , 3 , 4 , 4 = −8690 . 89 ;B 4 , 1 , 1 , 2 = 111 . 11 , B 4 , 2 , 2 , 2 = 456 . 603 , B 4 , 1 , 2 , 3 = 89 . 8146 , B 4 , 2 , 3 , 3 = 983 . 614 , B 4 , 1 , 1 , 4 = −981 . 733 ,

B 4 , 2 , 2 , 4 = −4412 . 05 , B 4 , 1 , 3 , 4 = −793 . 572 , B 4 , 3 , 3 , 4 = −8690 . 89 , B 4 , 2 , 4 , 4 = 5005 . 26 , B 4 , 4 , 4 , 4 = −14741 . 6 .

X. Wang et al. / Commun Nonlinear Sci Numer Simulat 65 (2018) 54–69 69

Appendix C

Table C1

The constant and varying matrices in the OFAPI algorithms.

Constant matrices Varying matrices

˜ E = I 2 L ×2 L � ( LB ) B −1 , ˜ J = J ( t ) , ˜ M = M � I M×M ,

˜ N = N (t ) ,

˜ C = C � I M×M , ˜ R =

E [ U 1

U 2 ] + [

−U 2

˜ M

−1 ˜ C U 2 +

˜ M

−1 ˜ N − ˜ F ]

t = [ t 1 , t 2 , ..., t M ] T

,

ˆ t = diag(t ) , ˜ t = I 2 L ×2 L � t , ˜ H = I 2 L ×2 L � ( L −1 B ) B −1 . ˜ F = F (t )

M is the number of collocation points in each time interval; L

is the length of variable vector x in Eq. (1) . � denotes the Kro-

necker product.

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