A Time Integration Scheme for Dynamic Problems GUWAHATI M.TECH THESIS.pdf · 2015-07-30 · Mishra,...

A Time Integration Scheme for Dynamic Problems

A Thesis Submitted

In Partial Fulfillment of the Requirements

for the Degree of

Master of Technology

by

Sandeep Kumar

Roll No. 134103123

to the

DEPARTMENT OF MECHANICAL ENGINEERING

INDIAN INSTITUTE OF TECHNOLOGY GUWAHATI

May, 2015

CERTIFICATE

It is certified that the work contained in the thesis entitled “A Time Integration Scheme for Dynamic

Problems”, by “ Mr. Sandeep Kumar” (Roll No. 134103123), has been carried out under my supervision

and that this work has not been submitted elsewhere for a degree.

Dr. S. S. Gautam

May, 2015. Department of Mechanical Engineering,

I.I.T. Guwahati.

Dedicated to

My Parents

and to

Dr. Pankaj Biswas and Dr. Rashmi Ranjan Das

Teachers and Friends

Acknowledgement

The most important lesson I have learned during the course of my work is that failures are part of life

and they are the best teachers who can guide one to success.

Any work of this stature has to have contributions of many people. During the course of this work,

I have been supported by many people. First of all, I would like to express my gratitude to thesis

supervisor, Dr. S. S. Gautam, for his guidance in completing the first phase of my project. The

technical and personal lessons that I learned by working under him are now foundation pillars for the

rest of my life.

I am specially grateful to Prof. Pankaj Biswas for his support, encouragement and inspiring advices

which will guide me all my life. I also extend my gratitude to Prof. Debabrata Chakraborty, Prof. A. K.

De, Prof. Karuna Kalita, Prof. Poonam kumari, Prof. G. Madhusudhana, Prof. K. S. R. Krishna Murthy,

Prof. Deepak Sharma and all other faculty members of the Department of Mechanical Engineering for

imparting me knowledge of various subjects and helping me at the time of difficulty in solving any

problem. I am grateful to Prof. Trupti Ranjan Mahapatra, Prof. Rashmi Ranjan Das, Prof. A. K. Sahoo

of KIIT University, Bhubaneswar and Prof. Subrata Panda of NIT Rourkela for their motivation and

support.

I am thankful to my parents, Shri A. Mohan Rao and Smt. A. Sarita, for providing me support and

encouragement at every step of my life. I am also thankful to my seniors, Dipendra Kumar Roy, Vinay

Mishra, Sibananda Mohanty, Manish Kumar Dubey, Sunil Kumar Singh, Debabrata Gayen, Susanta

Behera and Parag Kamal Talukdar. Further, I am also thankful to all my friends at IIT Guwahati -

Sandeep Kumar, Ashish Gajbhiye, Ashish Rajak, Nishiket Pandey, Soumya Ranjan Nanda, Anurag

Mishra, Nikhil Sharma, Abhishek Yadav, Sateesh Kumar, Vivek Badhe and Susobhan Patra. Finally, I

express my thanks to all those who have helped me directly or indirectly for successful completion of

this work.

Sandeep Kumar

IIT Guwahati

May, 2015

i

Conference Publications

• S. Kumar and S. S. Gautam, Extension of A Composite Time Integration Scheme for Dynamic Problems,

Indian National Conference on Applied Mechanics (INCAM 2015), July 13-15, 2015, New Delhi,

India, (accepted).

• S. Kumar and S. S. Gautam, Analysis of A Composite Time Integration Scheme, Indian National

Conference on Applied Mechanics (INCAM 2015), July 13-15, 2015, New Delhi, India, (accepted).

Contents

List of Figures xiii

List of Tables xv

Nomenclature xvi

1 Introduction 1

1.1 Need for Direct Time Integration Schemes . . . . . . . . . . . . . . . . . . . . . . . . . . . 1

1.1.1 Mode Superposition Method . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 1

1.1.2 Direct Time Integration . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 2

1.2 Objectives of the Thesis . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 3

1.3 Structure of the Thesis . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 3

2 Review of Direct Time Integration Schemes 4

2.1 Classification of Direct Time Integration Schemes . . . . . . . . . . . . . . . . . . . . . . 4

2.2 Classification of Collocation-Based Time Integration Schemes . . . . . . . . . . . . . . . 5

2.2.1 Explicit Time Integration Schemes . . . . . . . . . . . . . . . . . . . . . . . . . . . 6

2.2.2 Implicit Time Integration Schemes . . . . . . . . . . . . . . . . . . . . . . . . . . . 7

2.2.2.1 Literature Review on Implicit Time Integration Schemes . . . . . . . . . 8

2.2.2.2 Details of Some Implicit Time Integration Schemes . . . . . . . . . . . . 10

2.2.3 Selection of Explicit or Implicit Scheme . . . . . . . . . . . . . . . . . . . . . . . . 17

3 Proposed Time Integration Scheme 18

4 Analysis of Proposed Time Integration Scheme 22

4.1 Characteristics of Time Integration Schemes . . . . . . . . . . . . . . . . . . . . . . . . . . 22

4.1.1 Stability . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 22

4.1.2 Accuracy . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 24

4.1.2.1 Amplitude Error . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 25

4.1.2.2 Period Error . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 25

4.1.3 Damping . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 25

4.2 Stability and Accuracy Analysis of the Proposed Scheme . . . . . . . . . . . . . . . . . . 26

4.2.1 Amplification Matrix for the Proposed Scheme . . . . . . . . . . . . . . . . . . . . 26

5 Results and Discussion 35

5.1 Numerical example: Flexible Pendulum . . . . . . . . . . . . . . . . . . . . . . . . . . . . 35

5.2 Numerical example: Stiff Pendulum . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 37

xi

6 Conclusions and Scope for the Future Work 52

6.1 Summary . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 52

6.2 Scope of the Future Work . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 52

References 56

xii

List of Figures

3.1 Proposed Composite Scheme. The time step is denoted by tn+ 1 − tn = h. . . . . . . . . 18

4.1 Variation of spectral radii, amplitude error and period error for γt = 0.2. . . . . . . . . . 30





5.1 Flexible pendulum. Data and initial conditions. . . . . . . . . . . . . . . . . . . . . . . . 36

5.2 Variation of energy-momentum with time for h = 0.01 s and γt = 0.2. . . . . . . . . . . 37






5.8 Variation of energy-momentum with time for h = 0.0001 s and γt = 0.2. . . . . . . . . . 40



5.11 Variation of trajectory of the pendulum for h = 0.01 s. . . . . . . . . . . . . . . . . . . . . 41


5.13 Variation of trajectory of the pendulum for h = 0.0001 s. . . . . . . . . . . . . . . . . . . 43

5.14 Variation of strain with time for h = 0.0001 s. . . . . . . . . . . . . . . . . . . . . . . . . . 44

5.15 Variation of strain with time for h = 0.01 s. . . . . . . . . . . . . . . . . . . . . . . . . . . 45

5.16 Variation of energy-momentum with time for h = 0.1 s and γt = 0.2. . . . . . . . . . . . 45



xiii




5.22 Variation of axial strain with time for h = 0.0001 s. . . . . . . . . . . . . . . . . . . . . . . 48

5.23 Variation of axial strain with time for h = 0.1 s. . . . . . . . . . . . . . . . . . . . . . . . . 49

5.24 Variation of trajectory of the pendulum for h = 0.0001 s. . . . . . . . . . . . . . . . . . . 50


xiv

List of Tables

4.1 Newmark parameters. . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 29

xv

Nomenclature

Latin Symbols

h Time step size

u Displacement

u Velocity

u Acceleration

Greek Symbols

α Parameter for Chung and Hulbert (Generalised-α) scheme

αg Parameter for Gohlampour composite scheme

β Parameter for Newmark scheme

γ Parameter for Newmark scheme

γt Time step ratio for the proposed scheme

θ Parameter for Wilson-θ scheme

λ Eigenvalue

ξ Modal damping ratio

φ Mode shape

ω Vibration frequency

xvi

Chapter 1

Introduction

Transient response analysis is used to compute the dynamic response of a structure subjected to time-

varying excitation. The distinctive nature between static and dynamic problem is the presence of inertia

forces in dynamic problem which opposes the motion generated by the applied dynamic loading. The

dynamic nature of a problem is dominant if the inertia forces are large compared to the total applied

forces. When the motion generated by the applied forces are small such that the inertia forces are

negligible then the problem is considered as static [1].

Next, the need for time integration is detailed first in section 1.1 where two different approaches

to analyze the dynamic response of the material namely mode superposition method and direct time

integration schemes are discussed. Then, a detailed classification of direct time integration schemes is

presented in 2.1. Both the explicit and implicit schemes are discussed. Section 2.2.2 discusses various

implicit time integration schemes which are the focus of this research. The objective of the work are

discussed in 1.2. The chapter ends with section 1.3 which outlines the structure of the thesis.

1.1 Need for Direct Time Integration Schemes

In order to investigate the characteristics of transient dynamic problems, the resulting motion of

a structural dynamic problem is studied for a given load distribution in space and time. That is,

displacements, velocities and accelerations of degree of freedom as functions of time have to be studied.

There are two general approaches to analyze the dynamic response of structural systems namely (a)

Mode superposition method, and (b) Direct time integration schemes. Next, we briefly discuss each of

the two approaches.

1.1.1 Mode Superposition Method

Mode superposition method (also called the Modal Method) is a linear dynamic response procedure

which evaluates and superimposes free vibration mode shapes to characterize displacement patterns.

It determines the configurations into which the component displaces naturally. In any modal analysis,

1

only the lower frequencies and modes of the structure need to be retained. Modes retained must

have frequencies that span the temporal variation of loading. Mode shapes of free vibration are not

related to the complexity of the loading. The number of modes should be enough to approximate the

displacement associated with spatial variation of loading.

In modal superposition method, a set of uncoupled equations are obtained from the coupled

equations of motion of a discrete degree of freedom system using a transformation into modal or

normal coordinate space. In this space, each mode responds to its own mode shape φi, vibration

frequencyωi, and modal damping ξi. The total response can be obtained by summation of all the single

degree of freedom equations and hence, this method is known as modal superposition method [1].

Since the individual responses are superposed the only limitation of this method is that this method is

applicable for linear elastic systems.

For linear dynamic problems, where the response is dominated by low frequencies, mode super-

position method can be used to reduce the computational cost without sacrificing the accuracy. But

application of mode superposition method to nonlinear and real dynamic problems is very difficult,

leading to excessive computational costs. For such complex or nonlinear dynamic problems direct time

integration schemes prove to be more reliable and efficient.

1.1.2 Direct Time Integration

Modal methods use reduced set of degree of freedom to determine the displacements, velocities, and

accelerations as a function of time and then transform them back into original physical degree of

freedom space. On the hand in direct time integration scheme, no such transformation of equation of

motion is carried out. The response history is calculated using step-by-step integration in time. The

need for direct time integration is more when the equations of motion cannot be decoupled because

of a non-proportional damping matrix or because the system is nonlinear. These schemes are also

used to directly calculate the response of systems with large number of degree of freedoms to avoid

time-consuming calculations of eigenvalues and eigenvectors of the systems. Moreover, there is no

need to compute modes and frequencies in time integration scheme.

In direct time integration, the response history i.e., displacements, velocities, and accelerations are

calculated using step-by-step integration in time without changing the form of dynamic equations.

Equilibrium equations of motion are fully integrated as the structure is subjected to dynamic loading.

The governing equation of equilibrium for linear transient structural dynamic problems is expressed

as follows:

M u + C u + K u = F (t) (1.1)

where u is the displacement vector, M, C, and K are the mass, damping and stiffness matrices respec-

tively, and F is the vector of externally applied loads. The superimposed dots denotes derivative with

respect to time. In direct time integration, instead of satisfying Eq. (1.1) at any time t, the Eq. (1.1)

is satisfied only at discrete time intervals h apart. Therefore, equilibrium is achieved at discrete time

2

points within the interval of solution. Also, in direct time integration, a variation in displacements,

velocities, and accelerations is assumed within each time interval h. Stability, accuracy, and computa-

tional cost is dependent on the form of the assumption on the variation of displacements, velocities,

and accelerations within each time interval [2].

1.2 Objectives of the Thesis

The objective of the present work are as follows:

1. To propose an extension to an implicit time integration scheme of Silva and Bezerra [3]. It is

proposed to combine the Newmark scheme [4] with the three point backward Euler scheme to

have more user controlled numerical properties like high frequency dissipation.

2. To study the stability, accuracy, and dissipation of the proposed scheme. This is achieved by

studying the spectral radius, period elongation, and amplitude decay. The influence of the various

parameters like the Newmark parameters and substep size h on the stability and accuracy is also

carried out.

3. To study a number of linear and nonlinear dynamic problems using the proposed scheme.

1.3 Structure of the Thesis

The rest of the thesis is structured as follows. In Chapter 2, first the classification of time integration

schemes is presented. Both collocation-based and energy-momentum based schemes are described.

Then, more detailed classification of collocation-based schemes into explicit and implicit schemes is

presentee. Various schemes in each class are detailed. Finally, a detailed review of some recent implicit

time integration schemes is presented. The proposed time integration is discussed in Chapter 3.

Chapter 4 discusses the stability, accuracy, and dissipation of the proposed time integration scheme is

discussed. The performance of the proposed scheme is studied through various numerical examples

in Chapter 5. Chapter 6 concludes the thesis along with the scope of the future work.

3

Chapter 2

Review of Direct Time Integration

Schemes

There are many engineering problems in which dynamic effects play an important role. These problems

can be single body problems for e.g., civil engineering structures under environmental loads like wind,

water waves or earthquakes, in robotics. Also, they can be multi-body problems with contact-impact

as in for e.g., automobile crash simulation, design of landing gears for airplanes and space crafts, tire

wear simulation or adhesion simulation. These problems are mostly solved by direct time integration

of the equations of motion.

Direct time integration schemes are considered as the only general methods to calculate the response

of dynamic systems under any arbitrary loading. They are called direct because they are applicable

without modifications to the equations of motion of single degree of freedom and multi degrees of

freedom systems [1]. They determine the approximate values of the exact solution at discrete time

intervals. The principle of these methods can be summarized in two steps:

1. Assumption of some functions for time dependent variation of displacement, velocity and accel-

eration during a time interval h.

2. Satisfy the equation of motion at constant time interval h to maintain static equilibrium between

the inertia, the damping and the restoring forces and the applied dynamic loading at multiples

of the time step h.

2.1 Classification of Direct Time Integration Schemes

Traditionally, the direct time integration schemes for the nonlinear equation of motion have been

presented from two different perspectives: collocation-based schemes and momentum-based schemes [5].

In collocation-based schemes, the equation of motion is satisfied at selected points in the time

interval [tn, tn+1]. This gives one equation for the three variables: displacements (U), velocities (V = U)

and accelerations (A = U). Thus, two additional equations are needed. These are given by equations

4

relating the displacements, velocities and accelerations. In contrast, in the momentum-based schemes,

the equation of motion is developed over the time interval [tn, tn+1]. The idea is to integrate the equation

of motion over the respective time interval. Hence, while the inertial term becomes the finite increment

of momentum over the time interval, the external and the internal forces are represented by their time

averages [5]. The monograph by Wood [6] gives a detailed survey and mathematical background of

various implicit and explicit schemes developed until 1990. A third family of methods i.e., Galerkin

methods in time exist [7–9], but are not discussed further.

The Newmark scheme is one of the oldest and most popular collocation-based schemes which is

still extensively used [4]. It is known that even for the linear case, the Newmark scheme, or those

based on it, the energy is only conserved for a particular choice of the Newmark parameters β = 14

and γ = 12 . Even for this choice of parameters energy is not conserved for nonlinear systems. In many

applications only lower mode response is of interest. In such cases temporal integration schemes have

been developed with a controllable numerical dissipation for higher modes (see for e.g., [10–15]). Taking

γ > 12 and β >

γ2 in the Newmark scheme introduces so-called algorithmic damping into the Newmark

scheme. However, this also damps out the lower modes. A detailed analysis of energy conservation

and dissipation in linear Newmark-type algorithms and their αmodifications is discussed in [16].

It is well known that the traditional temporal discretization schemes like the Newmark based

schemes, which are unconditionally stable for linear problems, exhibit significant instabilities when

applied to nonlinear elastodynamics problems [17–19]. This has led to a significant amount of research

over the past two decades to develop more robust temporal discretization schemes for nonlinear

elastodynamic systems. A major focus has been to achieve numerical stability as well as maintain the

second order accuracy as in traditional methods. This has led to the development of the momentum

based schemes. These schemes have been developed with the idea of conserving properties of the

underlying problem like energy and momentum. Momentum based schemes have found their way

into elastodynamics through the pioneering work of Simo and Wong [20] and Simo and Tarnow [21].

They presented a new methodology for the construction of time integration algorithms that inherit, by

design, the conservation laws of momentum along with an a-priori estimate on the rate of decay of

the total energy. They called these algorithms energy momentum conserving algorithms (EMCA). The

proposed methodology considered a Saint Venant-Kirchhoff elasticity model. This scheme was further

extended to general elastic materials [22], systems with constraints [23, 24], shells [25, 26], composite

laminates [27] and multi-body dynamics [28].

2.2 Classification of Collocation-Based Time Integration Schemes

The collocation-based direct time integration schemes can be further classified into two types namely

explicit schemes and implicit schemes.

5

2.2.1 Explicit Time Integration Schemes

General form of difference equation for explicit scheme is expressed as

un+1 = f (un, un, un, un−1, un−1, . . .) (2.1)

In an explicit scheme, the displacements and velocities at the current time step tn+ 1 are found using

the values from the previous time step i.e., tn, tn− 1, tn− 2 and so on. The acceleration is then calculated

by substituting these values in Eq. 1.1 and solving system of simultaneous linear equations. In explicit

scheme, solver (direct or iterative) is not needed since the mass matrix is diagonalized and the variables

can be found simply dividing with force vector. In general, most explicit schemes are conditionally

stable and for nonlinear transient problems, nonlinear iterations within a time step is not required.

Hence its computer storage requirements is also less. It is to be ensured that time steps should be small.

For an explicit scheme, the results can be trusted to be reasonably accurate (for the given mesh size

under consideration), and typical time step studies as in implicit methods may not be justified [29].

Since cost per time step is small, explicit schemes are preferred in industry even though some trade

off is done with numerical accuracy. Explicit schemes [30] are suitable for wave propagation problems

where all modes participate in the solution. Some of the examples of explicit schemes are Central

difference scheme, Forward Euler scheme, Runge-Kutta scheme etc. The details of these schemes are

presented next.

(i) Central Difference Scheme: The central difference scheme is derived from the Taylor series [29] as

un =un+1 − un−1

2h−

h2

3

...un + . . . . (2.2)

This form of central difference scheme is for first-order ordinary differential equations to update

the approximation to the solution at the time t = tn+1 in terms of the approximation to the solution

at the previous step time t = tn−1. Approximation to the solution un+1 at the time t = tn+1 is

given as

un+1 = un−1 + 2 h un + O(h3) , (2.3)

where O(h3) is the local truncation error. The central difference formula for velocity is given by

un+ 12=

1

h( un+1 − un ) , (2.4)

and the acceleration is given by

un =1

h2( un+1 − 2 un + un−1 ). (2.5)

(ii) Runge-Kutta Scheme: The second order Runge-Kutta scheme is a one step explicit scheme. The

approximation to the solution at the time t = tn+1 for the first-order linear ordinary differential

6

equation is given as [29]

un+1 = un +

∫ tn+1

tn

f (u, t)dt. (2.6)

The second order Runge-Kutta scheme is given as [29]

un+1 = un + k2. (2.7)

k1 = h f (un , tn).

k2 = h f(

un +k1

2, tn+ 1

2

)

.

The second-order Runge-Kutta scheme has third-order local truncation error O(h3). The fourth-

order Runge-Kutta scheme is one-step explicit scheme [29] and is given as

un+1 = un +1

6(k1 + 2 k2 + 2 k3 + k4) , (2.8)

k1 = h f (un , tn) , (2.9)

k2 = h f(

un +k1

2, tn+ 1

2

)

,

k3 = h f(

un +k2

2, tn+ 1

2

)

,

k2 = h f (un + k3 , tn+ 12).

The fourth-order Runge-Kutta scheme has the local truncation error, O(h5).

(iii) Forward Euler Scheme: The forward Euler scheme is a one-step explicit scheme. This scheme can

be obtained by a Taylor series of order one around the point tn. The approximation for velocity

at time tn is given as

un =un+1 − un

h. (2.10)

Since the scheme is first-order accurate in time, it is called first-order scheme. Approximation to

the solution un+1 is given follows

un+1 = un + h un = un + h f (un, tn) (2.11)

2.2.2 Implicit Time Integration Schemes

General form of difference equation for an implicit scheme is expressed as [30]

un+1 = f (un, un, un, un−1, un+1, . . .) . (2.12)

In the implicit schemes, the displacements and the velocities at the current time step are expressed

not only in terms of the values of the previous time step but also of the current time step. Hence,

the solution of system of resulting equations requires an iterative scheme, usually Newton-Rapshon

method, to obtain the solution. This allows for larger time step size to be used during the analysis.

Also, the cost per time step is greater and requiring more computer storage space compared to explicit

7

method. Implicit schemes are suitable for structural dynamics problems (inertial or vibrations type

applications) where mostly the low frequency modes are dominant. The implicit trapezoidal scheme

is unconditionally stable for the linear dynamic problems. However, time step studies are required for

implicit schemes as the accuracy of the result is not guaranteed at any arbitrary time step value [30].

Next, a detailed literature review of some recent implicit time integration schemes is presented.

2.2.2.1 Literature Review on Implicit Time Integration Schemes

The Newmark scheme is one of the oldest and most popular collocation-based schemes which is still

extensively used [4]. It is known that even for the linear case, the Newmark scheme, or those based

on it, the energy is only conserved for a particular choice of the Newmark parameters β = 14 and

γ = 12 . Even for this choice of parameters energy is not conserved for nonlinear systems. In many

applications only lower mode response is of interest. In such cases temporal integration schemes have

been developed with a controllable numerical dissipation for higher modes (see for e.g., [10–15]). Taking

γ > 12 and β >

γ2 in the Newmark scheme introduces so-called algorithmic damping into the Newmark

scheme. However, this also damps out the lower modes. A detailed analysis of energy conservation

and dissipation in linear Newmark-type algorithms and their αmodifications is discussed in [16].

A comprehensive study on direct time integration schemes have been done by Subbaraj and Dokain-

ish [31] and Bert [32]. They have done comparative evaluation of different time integration schemes

along with their implementation to some numerical problems. Another important characteristic of time

integration schemes i.e., overshooting, have been studied by Hilber [33]. Also, along with overshooting

characteristic, an elaborate study of collocation time integration schemes have been done by Hilber [33].

Stability region for time integration schemes has been studied by Park [34]. He has also made a de-

tailed study of stiffly stable methods. Benitez and Montans [35] have obtained the amplification matrix

numerically and discussed the overshooting effects. This is a powerful method to check whether the

algorithm has been initialized correctly according to real initial conditions of the problem or not.

Several other time integration schemes have been developed with an aim to improve the charac-

teristics of time integration schemes. Hilber and et al. [10] have developed a time integration scheme

popularly known as HHT-α scheme. This scheme is unconditionally stable and it has been developed

for better preservation of low frequency modes. Another scheme which is a modification of Newmark

scheme has been developed by Wood et al. [11] and is popularly known as WBZ-α scheme. Chung and

Hulbert [12] have combined Newmark, HHT-α, and WBZ-α schemes and developed a new scheme

popularly known as Generalized-α scheme. This scheme is second order accurate and unconditionally

stable. They also studied the stability and accuracy characteristics of the proposed scheme. Zhou and

Zhou [36] proposed an implicit time integration scheme which has two control parameters to vary

the accuracy. To capture the high oscillatory modes accurately Liang [37] proposed a time integration

scheme where acceleration within a particular time step is assumed to vary in a sinusoidal manner.

Gholampour et al. [38–40] have proposed an unconditionally stable time integration scheme in which

8

order of acceleration has been increased by including more terms of the Taylor series. Stability, accuracy,

and overshooting characteristics of the scheme was studied. The performance of the proposed scheme

is compared with other time integration schemes by applying it to some linear and nonlinear exam-

ples. In another time integration scheme by Gholampour and Ghassemieh [41], the approximation for

displacement term is considered as fourth order polynomial with five coefficients. They studied the

characteristics of the proposed scheme for different damping and stiffness ratios. Weighted residual

integration is used for determination of these coefficients. Celay and Anza [42] proposed a linear mul-

tistep method known as BDF-α, where parameter α controls the numerical dissipation and stability.

Alamatian [43] discussed a new multistep time integration (N-IHOA). Displacement and velocity vec-

tors at current time step are proposed to be functions of velocities and accelerations of several previous

time steps respectively. As several acceleration and velocity terms are included in the approximation

for displacement and velocity effects of local and residual errors are reduced. Chang [44] discussed a

new family of structure-dependant methods (SDM-2) and compared it with SDM-1. No overshoot in

displacement and velocity for SDM-2 is observed. Also, SDM-2 was found to be computationally more

efficient than SDM-1.

A collocation based composite time integration method has been proposed by Bathe and co-

workers [45, 46]. The scheme is usually referred as Bathe composite scheme. The idea is to combine

a highly dissipative time integration scheme with a non-dissipative time integration scheme. The

method combines the trapezoidal rule and the three-point backward Euler scheme to yield a composite

scheme for numerical integration of nonlinear dynamical system of equations. The method, unlike for

e.g., Newmark scheme, has no parameter to choose or adjust. The method is shown to be second order

accurate and remains stable for large deformation and long time response. The time integration scheme

is simple and computationally efficient within the Newton-Raphson iterations. However, this method

does not directly impose energy and momentum conservation. Dong [47] has presented various time

integration algorithms of second order accuracy based on a general four-step scheme that resembles

the backward differentiation formulas. An extension to the composite strategy of Bathe [45, 46] is

proposed.

Recently, Silva and Bezerra [3] have proposed a scheme which is based on the Bathe campsite

scheme [46] but with generalised substep sizes instead of equal substep size as used in the Bathe com-

posite scheme. The algorithm preserves energy-momentum without the need for Lagrange multipliers

in the scheme for energy and momentum conservation. They have shown that for too large time step,

the scheme remains stable but numerical dissipations are also large. Klarmann and Wagner [48] have

further analyzed the Bathe composite scheme for variable step sizes and have shown that at a particular

value of the step size the period elongation is minimum and the numerical dissipation is maximum.

9

2.2.2.2 Details of Some Implicit Time Integration Schemes

In the current section, details of some implicit time integration schemes is presented. The proposed

scheme is discussed in detail in Chapter 3.

(i) Backward Euler Scheme: The backward Euler scheme is a one-step implicit scheme, where the

numerical integration scheme prescribes updating of the approximation to the solution at the

time t = tn+ 1 in terms of the approximation to the solution at the current step time t = tn+ 1 [29].

The first-order backward Euler scheme is given by

un+ 1 =un+ 1 − un

h. (2.13)

(ii) Newmark Scheme: Newmark Scheme [4] is the most widely used family of direct time integration

schemes. This scheme can be used as a single-step or a multi-step algorithm. For a single-step

three-stage algorithm, un, un, and un have to be calculated at each time step. For a single-step

two-stage algorithm, un, and un have to be calculated at each time step. It can be also used as a

predictor-corrector form [6]. Using dynamic equilibrium equations at time level tn+1 (Eq. 1.1), we

have :

M un+1 + C un+1 + K un+1 = Fn+1 . (2.14)

The approximations for displacement and velocity at time tn+1 for Newmark scheme are given by

un+1 = un + h un +1

2h2 [(1 − 2 β)un + 2 β un+ 1]. (2.15)

un+ 1 = un + h [(1 − γ) un + γ un+ 1]. (2.16)

where β and γ are called the Newmark parameters [4]. These parameters indicate how much of

the acceleration at the end of the time step enter into the relation for velocity and displacement

at the end of the time step [6]. From these relations, the unknowns un+1, un+1 and un+1 are

determined from the known values of un, un and un.

Remarks:

• The conditions for the unconditional stability of Newmark scheme are

2 β ≥ γ ≥1

2. (2.17)

When γ > 12 and β >

γ2 in the Newmark scheme, algorithmic damping is introduced [29].

• For different values of β and γ different schemes are obtained. Some are given below.

Constant(average) acceleration scheme: For β = 14 and γ = 1

2 , Newmark scheme gives

average acceleration scheme, which is implicit, second order-accurate and unconditionally

stable.

10

Linear acceleration scheme: For β = 16 and γ = 1

2 , Newmark scheme yields linear acceler-

ation scheme, which is implicit and conditionally stable.

Fox-Goodwin scheme: For β = 112 and γ = 1

2 , Newmark scheme gives Fox-Goodwin

scheme, which is implicit and conditionally stable. In absence of viscous damping, this

scheme is fourth-order accurate.

Central difference scheme: For β = 0 and γ = 12 , Newmark scheme gives central dif-

ference scheme, which is conditionally stable. When M and C are diagonal, the scheme

is explicit. Central difference scheme is generally the most economical direct integration

scheme and widely used when the time step restriction is not too severe, such as in elastic

wave propagation problems.

(iii) Wilson-θ Scheme: Wilson-θ scheme is an extension of the linear acceleration scheme. It is an

implicit scheme and does not require any special starting procedures. This scheme is uncon-

ditionally stable. Linear variation of acceleration is assumed over the time interval from tn to

tn + θ h, where θ ≥ 1. Using dynamic equilibrium equations at time level t + θ h (Eq. 1.1), we

have

M un+θ + C un+θ + K un+θ = Fn+θ , (2.18)

where

Fn+θ = Fn + θ (Fn+ 1 − Fn). (2.19)

Here, θ is a free parameter which controls the stability and accuracy of the algorithm [29]. The

approximations for displacement and velocity can be written as

un+θ =6

θ2 h2(un+θ + un) −

6

θun − 2 un. (2.20)

un+θ =3

θ h(un+θ + un) − 2 un −

θ h

2un.

Remarks: Wilson-θ reduces to linear acceleration scheme for θ = 1 but for parameters θ ≥ 1.37,

the scheme is unconditionally stable. For values θ > 2, numerical dissipation reduces and

relative period error increases. Also, high overshooting behavior is a disadvantage in Wilson-θ

scheme [29].

(iv) Houbolt Scheme: Houbolt scheme is an implicit, unconditionally stable, and second order accurate

scheme [29]. It is not self-starting. It requires special starting procedure for the determination

of the displacements at previous time steps. In order to find the approximation for velocity

and acceleration at tn+1, Houbolt scheme uses equation of motion at tn+1 and two backward

difference formulae, which is obtained from a cubic polynomial passing through four successive

time levels [29]. Using dynamic equilibrium equations at time level tn+ 1 (Eq. 1.1), we have

M un+ 1 + C un+ 1 + K un+ 1 = Fn+ 1. (2.21)

11

The approximations for velocity and acceleration at time tn+1 is given by

un+1 =1

6h(11 un+1 − 18 un + 9 un−1 − 2 un−2). (2.22)

un+1 =1

h2(2 un+1 − 5 un + 4 un−1 − un−2).

Remarks: Houbolt scheme has higher damping and relative period error compared to other

schemes. An important disadvantage of Houbolt scheme is, it has excessive algorithmic damping

and affect the low frequency modes too strongly. Also, there is no parametric control over

algorithmic damping.

(v) HHT-α Scheme: The Hilber-Hughes-Taylor-α scheme (HHT-α) is unconditionally stable, second

order accurate, possesses high frequency numerical dissipation which can be controlled by free

parameter α rather than by the time step size so that it does not affect the lower modes too

strongly. Newmark scheme [4] is used as a basis and as a starting point for this scheme. Equations

(2.16 - 2.15) are used for approximations of displacement and velocity. The dynamic equilibrium

equation (Eq. 1.1) is written in a modified form as

M un+ 1 + (1 + α) C un+ 1 − αC un + (1 + α) K un+ 1 − K un = Ftn +α. (2.23)

where the parameter α is used to vary the numerical dissipation of the scheme, and tn+ 1+α =

(1 + α) tn+ 1 − α tn = tn+ 1 + α h ,

Remarks: The conditions for unconditional stability of HHT - α scheme are

−1

3≤ α ≤ 0 , (2.24)

β =(1 − α)2

4, (2.25)

γ =1 − 2α

2. (2.26)

The parameter α also governs the numerical dissipation of the algorithm where larger negative

values of α signifies the increase in amount of amplitude and period error. Small negative values

of α will have the opposite effect. For α = 0 the algorithm reduces to implicit Newmark scheme

(γ = 12 , β =

12 ). Amount of numerical dissipation and relative period error is less compared

to Houbolt and Wilson-θ schemes. HHT - α is a U0-V1 scheme i.e., zero-order overshoot in

displacement and first-order overshoot in velocity [29].

(v) Wood-Bosak-Zienkiewicz (WBZ) Scheme: This scheme is based on HHT-α with controllable dis-

sipation parameter [11]. This scheme retains the Newmark’s scheme approximation equations

for displacement and velocity i.e., Eqs.( 2.16- 2.15). Using dynamic equilibrium equations at time

12

level tn+ 1 (Eq. 1.1), we have

(1 − αB) M un+ 1 + αB M un + C un+ 1 + K un+ 1 = Fn+ 1. (2.27)

where αB is the algorithmic parameter.

Remarks:

• For αB = 0, WBZ scheme reduces to Newmark average acceleration scheme.

• The scheme is second-order accurate in time and unconditionally stable for the following

conditions

αB = 0 , (2.28)

γ =1

2− αB , (2.29)

β =1

4(1 − αB)2 . (2.30)

(vi) Chung and Hulbert Scheme (Generalized-α): Generalized-α is a combination of HHT-α and WBZ

schemes. The dynamic equilibrium equations (Eq. 1.1) is written in modified manner as

Mun+ 1−αm+ C un+ 1−α f

+ K un+ 1−α f= Fn+ 1−α f

, (2.31)

where

un+ 1−α f= (1 − α f ) un+ 1 + α f un , (2.32)

un+ 1−α f= (1 − α f ) un+ 1 + α f un , (2.33)

un+ 1−αm= (1 − αm) un+ 1 + αm un , (2.34)

tn+ 1−α f= (1 − α f ) tn+ 1 + α f tn . (2.35)

Remarks:

• For αm = α f = 0, the scheme reduces to Newmark scheme [4].

• For αm = 0 andα f = 0, the scheme reduces to HHT-α scheme and WBZ scheme respectively.

• Conditions for unconditional stability are

αm ≤ α f ≤1

2, (2.36)

β ≥1

4+

1

2(α f − αm) . (2.37)

The scheme is second order accurate for the following condition

γ =1

2− αm + α f . (2.38)

13

(vii) Bathe Composite Scheme: In the Bathe composite scheme [45, 46, 49], a highly dissipative time

integration scheme is combined with a non-dissipative time integration scheme. For conservation

of energy and momentum, trapezoidal scheme is combined to three-point backward Euler scheme.

Trapezoidal scheme ensures second order accuracy and the three-point backward Euler scheme

ensures high-frequency numerical dissipation. One time step h is subdivided into two substeps

of sizes h/2 each. For the first substep, trapezoidal scheme is used. Then, for the second substep,

three-point backward Euler scheme is used. Using dynamic equilibrium equations at time level

tn+ 12

(Eq. 1.1), we have :

M un+ 12+ C un+ 1

2+ K un+ 1

2= Fn+ 1

2. (2.39)

Considering tn+ 12= tn +

h2 , Trapezoidal scheme is applied over the first substep. The approxima-

tions for velocity and displacement at time tn+ 12

for Trapezoidal scheme are given by

un+ 12= un +

h

4(un + un+ 1

2). (2.40)

un+ 12= un +

h

4(un + un+ 1

2).

In the second substep, the dynamic equilibrium equation are written at time level tn+ 1 (Eq. 1.1)

as

M un+ 1 + C un+ 1 + K un+ 1 = Fn+ 1. (2.41)

Considering tn+ 1 = tn + h, three-point backward Euler scheme is applied over the second substep.

The approximations for velocity and displacement at time tn+ 1 for three-point backward Euler

scheme are given by

un+ 1 =1

hun −

4

hun+ 1

2+

3

hun+ 1 , (2.42)

un+ 1 =1

hun −

4

hun+ 1

2+

3

hun+ 1 . (2.43)

(viii) Composite Scheme of Silva and Bezerra: Silva and Bezerra [3] proposed a composite scheme

which is based on the Bathe composite scheme [45, 46, 49] but with generalised substep sizes

instead of equal substep size used in the Bathe composite scheme. Considering tn+γt= tn + γt h

as an instance of time between tn and tn+1 for γt ∈ (0, 1), trapezoidal scheme is applied over the

first substep, γt h. Using dynamic equilibrium equations at time level tn+γt(Eq. 1.1), we have

M un+γt+ C un+γt

+ N (u , tn+γt) = Fn+γt

, (2.44)

where M is the mass matrix, C is the damping matrix, N (u , tn+γt) is the internal force vector

which is, in general, a function of displacement vector u and time t, and Fn+γtis the external

force vector. The vectors of velocity and acceleration are represented by u, and u respectively.

Note that for linear dynamic analysis, the internal force vector N (u , tn+γt) can be written as K u

14

where K is the stiffness matrix. The approximations for velocity and displacement for trapezoidal

scheme are given by

un+γt= un +

1

2(un + un+γt

)γt h , (2.45)

un+γt= un +

1

2(un + un+γt

)γt h . (2.46)

Considering tn+ 1 = tn + (1 − γt) h as an instance of time between tn and tn+1 for γt ∈ (0, 1),

three-point backward Euler scheme is applied over the second substep, (1 − γt) h. Using dynamic

equilibrium equations at time level tn+ 1 (Eq. 1.1), we have

M un+ 1 + C un+1 + N (u , tn+ 1) = Fn+ 1 . (2.47)

The approximations for velocity and acceleration for three-point backward Euler scheme are

given by

un+ 1 = c1 un + c2 un+γt+ c3 un+ 1 , (2.48)

un+ 1 = c1 un + c2 un+γt+ c3 un+ 1 , (2.49)

where the constants c1, c2, and c3 can be expressed as

c1 =(1 − γt)

γt h, (2.50)

c2 =−1

(1 − γt)γt h, (2.51)

c3 =(2 − γt)

(1 − γt) h. (2.52)

(ix) Composite Scheme by Gohlampour et al.: Recently a new scheme has been proposed by Gohlam-

pour et al. [38–40] for direct time integration for non-linear dynamic problems. In order to improve

the accuracy of the composite scheme, the order of acceleration was increased by including more

terms of the Taylor series. Two parameters αg and δ control the accuracy of the scheme. This is

a two-step integration scheme as the responses at ’t + 1’ depend on responses at ’t’ and ’t - 1’.

Using Taylor series expansion, approximations for displacement and velocity for this scheme are

given by

ut+ h = ut + h ut +h2

2ut +

h3

6

...ut + αg h4 ....

u t , (2.53)

ut+ h = ut + h ut +h2

2

...ut + δ

h3

6

....u t . (2.54)

15

The value of the vectors...ut and

....u t are computed using following approximation

...u t =

1

2 h(ut+ h − ut− h) (2.55)

....u t =

1

h2(ut+ h + ut− h − 2 ut)

Remarks:

• The composite scheme by Gohlampour is unconditionally stable for the following values of

δ and αg

δ ≥1

3(2.56)

δ

2≤ αg ≤ δ −

1

6. (2.57)

• The Gohlampour scheme maintains second order accuracy while numerical damping in

contrast to Newmark scheme [4] where numerical damping can be produced but with first

order accuracy.

• Relative period error in this scheme is similar compared to Newmark’s average acceleration

and Generalised-α schemes but lesser than Wilson-θ scheme.

(x) Park Scheme: Park scheme is a linear three-step scheme [29]. It is an implicit, second-order accurate

and unconditionally stable scheme. This scheme is similar to Houbolt scheme. The dynamic

equilibrium equation is same as Eq. (2.21). Approximations for velocity and acceleration, however,

at time tn+ 1 are given as

un+1 =1

6h(10 un+1 − 15 un + 6 un−1 − un−2) , (2.58)

un+1 =1

6h(10 un+1 − 15 un + 6 un−1 − un−2) . (2.59)

Remarks:

• Park scheme is a stiffly stable scheme. This method can also be formulated from Gear’s

two-step and three-step stiffly stable schemes.

• Using Park’s scheme for a stiff system equation of motion can be integrated with a large time

step size.

• It is strongly dissipative in the high frequency region similar to Houbolt scheme. It has

improved characteristics for amplitude error and period error as compared to Houbolt

scheme [29].

16

2.2.3 Selection of Explicit or Implicit Scheme

In practical application, the choice between an implicit and explicit schemes are on the basis of stability

and economy. A prominent disadvantage of the explicit schemes is that they are only conditionally

stable. This means that the time step size has to be below a critical value hcr. If large time steps (greater

than the critical time step hcr) are used, the numerical solution blows off and it diverges completely

from the actual solution. Explicit schemes are widely used for fast transient analysis, for example, in

the analysis of crash problems.

On the other hand in the implicit schemes, the displacement and the velocity at the current time

step are expressed not only interms of the values of the previous time step but also of the current

time step. Hence, the solution of system of resulting equations requires an iterative scheme, usually

Newton-Rapshon method, to obtain the solution. This allows for larger time step size to be used during

the analysis. As such, there is no such restriction on size of time step h.

17

Chapter 3

Proposed Time Integration Scheme

In the present chapter, an extension to the composite scheme proposed by Silva and Bezerra [3] is

preseneted. In the proposed extension too the variable substep sizes are used. However, the proposed

implicit composite scheme is a parameter based time integration scheme in which the Newmark

scheme [4] is applied in the first substep and three-point backward Euler scheme for the second

substep. The composite scheme is shown schematically in Figure (3.1).

Figure 3.1: Proposed Composite Scheme. The time step is denoted by tn+ 1 − tn = h.

The governing equations of equilibrium for nonlinear transient structural dynamic problems is ex-

pressed as follows:

M u + C u + N(u, t) = F(t) (3.1)

where M is the mass matrix, C is the damping matrix, N(u, t) is the internal force vector which is, in

general, a function of displacement vector u and time t and F(t) is the external force vector. The vectors

of velocity and acceleration are represented by u, and u respectively. Note that for linear dynamic

analysis, the internal force vector N(u, t) can be written as K u where K is the stiffness matrix. Next,

the proposed scheme is explained in detail by applying it to Eq.( 3.1).

Considering tn+γt= tn + γt h (where h is the time step size) as an instance of time between tn and tn+1

for γt ∈ (0, 1), Newmark scheme is applied over the first substep, γt h (see Fig. 3.1). The approximations

18

for displacement and velocity at time tn+γtfor Newmark scheme are given by

un+γt= un + γt h [ (1 − γ) un + γ un+γt

] , (3.2)

un+γt= un + (γt h) un +

(γt h)2

2

[

(1 − 2 β) un + ( 2 β ) un+γt

]

, (3.3)

where β, γ are Newmark scheme parameters. After rearrangement, the acceleration and velocities at

time tn+γtcan be written as

un+γt=

1

β (γt h)2( un+γt

− un) −1

β (γt h)un −

(

1

2 β− 1

)

un , (3.4)

un+γt=

γ

β (γt h)( un+γt

− un) +

(

1 −γ

β

)

un + (γt h )

[

(1 − γ) − γ( 1

2 β− 1

)

]

un . (3.5)

The equilibrium equation given by Eq. (3.1) is written at time tn+γtas

M un+γt+ C un+γt

+ N (u , tn+γt) = Fn+γt

. (3.6)

Substituting for the expression for acceleration from Eq. (3.4), Eq. (3.6) can be written as

M

[

1

β (γt h)2( un+γt

− un) −1

β (γt h)un −

( 1

2 β− 1

)

un

]

+ N (u , t)n+γt= Fn+γt

(3.7)

Then, the residual is defined as

Rn+γt= M

[

1

β (γt h)2( un+γt

− un) −1

β (γt h)un −

( 1

2 β− 1

)

un

]

+ N (u , t)n+γt− Fn+γt

(3.8)

This equation is solved by consistent linearization and Newton-Raphson method [50] and effective

stiffness matrix K (uin+γt

) at ith iteration is obtained which is deformation-dependent. The expression

for K (uin+γt

) is given by

K (uin+γt

) =1

β (γt h)2M + Kt (un+γt

) (3.9)

where

Kt (un+γt) =

∂N (u , t)n+γt

∂un+γt

. (3.10)

The matrix Kt (un+γt) is also called the algorithmic tangential stiffness matrix. The effective iterative

equation is given by

K (uin+γt

)∆ui= −Rn+γt

. (3.11)

The displacements are updated as

ui+ 1n+γt

= uin+γt

+ ∆ui. (3.12)

In the second substep the three point backward Euler scheme is applied over the second substep

( 1 − γt ) h. The approximation for velocity and acceleration at time tn+ 1 for the three point backward

19

Euler scheme is given by

un+ 1 = c1 un + c2 un+γt+ c3 un+ 1 , (3.13)

un+ 1 = c1 un + c2 un+γt+ c3 un+ 1 ,

where the constants are given as

c1 =(1 − γt)

γt h, (3.14)

c2 =−1

(1 − γt)γt h, (3.15)

c3 =(2 − γt)

(1 − γt) h. (3.16)

Again substituting for the vecolity at time tn+1 from Eq. (3.13) in the expression for acceleration in

Eq. (3.14) we obtain

un+ 1 = c1 un + c2 un+γt+ c3 c1 un + c3 c2 un+γt

+ c23 un+ 1 . (3.17)

The equilibrium equation given by Eq. (3.1) is now written at time tn+ 1 as

M un+ 1 + C un+ 1 + N (u , t)n+ 1 = Fn+ 1. (3.18)

Substituting for acceleration from Eq. (3.17) we obtain

M

[

c1 un + c2 un+γt+ c3 c1 un + c3 c2 un+γt

+ c23 un+ 1

]

+ N (u , t)n+ 1 = Fn+ 1 (3.19)

Then, the residual is defined as

Rn+ 1 = M

[

c1 un + c2 un+γt+ c3 c1 un + c3 c2 un+γt

+ c23 un+ 1

]

+ N (u , t)n+ 1 − Fn+ 1 (3.20)

Again, this equation is solved by consistent linearization and Newton-Raphson iterative method and

effective stiffness matrix K (uin+ 1

) at iteration i is obtained which is deformation-dependent. The

expression for K (uin+ 1

) is given by

K (uin+ 1) = c2

3 M + Kt (un+ 1) , (3.21)

where

Kt (un+γt) =

∂Nn+ 1

∂un+ 1. (3.22)

The effective iterative equation is given by

K (uin+ 1)∆ui

= −Rn+ 1 . (3.23)

20

The displacements are updated as

ui+ 1n+ 1 = ui

n+ 1 + ∆ui. (3.24)

21

Chapter 4

Analysis of Proposed Time Integration

Scheme

In the present chapter, first, some properties that a time integration scheme should possess i.e., stability,

accuracy, and high frequency damping are discussed. Then, the stability and accuracy characteristics

of the proposed scheme, presented in Chapter 3, is studied for various values of parameters.

4.1 Characteristics of Time Integration Schemes

The trapezoidal scheme is unconditionally stable for the linear dynamic problems. However, for

nonlinear dynamic problems, the trapezoidal scheme does not guarantee the conservation of energy

and momentum as time progresses [18, 26, 45, 51]. It fails to provide high frequency dissipation in

nonlinear analysis. Even if smaller time step is considered convergence is not guaranteed as it may

lead to excitation of even higher frequencies which lead to instability. One of the earliest work on the

spectral stability and accuracy analysis of direct time integration schemes has been done by Bathe and

Wilson [52]. Also, Bathe [2] has discussed the stability and accuracy characteristics of several direct time

integration schemes (both implicit and explicit time integration schemes). In linear dynamic analysis,

the spectral stability is sufficient condition for unconditional stability of the time integration scheme

[53]. However, for nonlinear dynamic analysis spectral stability is required but it is only a necessary

condition [26]. Numerical dissipation is considered to be advantageous as it ensures better numerical

stability for time integration schemes.

4.1.1 Stability

Stability can be loosely defined as the property of an integration method to keep the errors in the

integration process of a given equation bounded at subsequent time steps. For dynamic problems,

when finite differences or finite elements are used to discretize the spatial domain, the spatial resolution

of high-frequency modes are poor [54]. Numerical high frequency modes are artificially introduced.

22

To improve the convergence of iterative equation solvers for nonlinear problems, algorithmic damping

is included in a step-by-step time integration scheme. Algorithmic damping helps to preserve the

low frequency modes and damping out high frequency modes in a controlled way. It also helps in

solving problems which involve constraints, for example, contact problems [26]. Algorithms which

are unconditionally stable for linear dynamic systems often loose their stability in nonlinear problems

and this makes spectral stability a necessary condition for stability of time integration schemes for

nonlinear problems. Broadly, stability can be analyzed by two methods. One is spectral or Fourier

stability analysis, which examines the equations of motion of a single degree of freedom. The second

method is energy stability analysis, where equations of original system are analyzed and conditions are

established such that as time increases, norm of the solution remains bounded [30].

Spectral stability is concerned with the rate of growth, or decay of powers of the amplification

matrix. In spectral stability, the dissipation can be measured by spectral radius ρ ( A) and it is the

largest magnitude of the eigenvalues of the numerical amplification matrix, A. According to Wood [6],

the spectral radius should stay close to unity level as long as possible and it should decrease to about

0.5 − 0.8 as hT (T is the undamped natural period) tends to infinity. When h

T → ∞, the corresponding

spectral radius is known as the ultimate spectral radius. The property in which the ultimate spectral

radius approaches zero and the high-frequency responses are eliminated in one step, is known as

asymptotic annihilation. Hence the conditions for spectral stability can be summarized as [55] :

1. The spectral radius ρ ( A), which is the maximum of the eigen values of amplification matrix, A,

should be less than or equal to one i.e., ρ ( A) ≤ 1.

2. Eigenvalues of A of multiplicity greater than one, are strictly less than one in modulus.

A matrix A which satisfies both the above conditions is said to be (spectrally)stable. Here only three

kinds of stability which are of use for the analysis of multibody systems are addressed.

• Conditional and unconditional stability-Algorithms that are stable for some restricted range of

values (λ h) (area of the complex plane) are called conditionally stable. In case of unconditionally

stable algorithms, there is no restriction on the size of step size (h). For conditionally stable

algorithms, the time step should be below a critical value. This value depends on the charac-

teristics of the problem which is defined by the eigen value (λ) (or a set of (λ). In order to find

the range of values (λ h) in the complex plane for which the scheme is stable, region of absolute

stability is defined for a scheme. The region of absolute stability is an intrinsic characteristic of

a scheme which should be considered while using conditionally stable algorithms. Allowing λ

to be complex comes from the fact that in practice we are usually solving a system of ordinary

differential equations (ODEs). In the linear case stability is determined by the eigenvalues of the

coefficient matrix. In the nonlinear case we typically linearize and consider the eigenvalues of

the Jacobian matrix. The numerical stability of a time-integration scheme is related to its spectral

stability. Numerical instability during time integration may occur if the spectral radius exceed

23

unity for some hT , hence increasing the error exponentially [54]. For unconditional stability, spec-

tral radius should be less than unity for all hT i.e. any time step size,h, can be used. Large time

step size is advantageous in dynamic problems where responses are primarily contributed by the

low-frequency modes.

• Stiffly stable- Stiffly stable [55] method is one which is absolutely stable in the region of λ h-plane

and defined by Re (λ h ) < − δ, where δ is a positive constant.During numerical integration of

a differential equation, the step size is usually small where the variation of the solution curve

is more and the step size is taken relatively large where the slope of the solution curve nearly

approaches zero. Sometimes, during numerical integration, when the step size is small even

when the solution curve is smooth, the system is considered to be stiff. Hence this phenomenon

is known as stiffness. The integration of these systems by conditionally stable algorithms should

be avoided, because it would require small time steps and hence making it computationally

expensive and inaccurate solutions due to round-off errors.

• A-stable- An algorithm is said to be A-stable if the solution to u= λu tends to zero as n→ ∞when

the Re (λ) < 0, which means that the numerical solution decays to zero when the corresponding

exact solution decays to zero. Multi-body systems may have pure vibration modes whose eigen

values may lie in the imaginary axis. Stiff stable methods are inadequate, whose region of

absolute stability do not include imaginary axis of the complex plane. Hence A-stable methods

are required for such problems. A-stable algorithms are considered to be unconditionally stable

for linear problems as there is no limitation on the size of h for the stability of the integration

process. An important sub-class of A-stable methods is L-stable. The difference between L-

stability and A-stability is that the former damps out the response of stiff components (equations

with high eigen value, λ ) very rapidly, almost in only one time step. This property is applicable

only in those cases with spurious stiff equations which arise during the formulation or modeling

process.

4.1.2 Accuracy

Accuracy is determined by amplitude error and period elongation introduced by the numerical integration

scheme in comparison with exact response for the conservative system (ξ=0) during free vibrations

(F=0). The equation of motion for the conservative system with free vibrations is given as

u + ω2 u = 0. (4.1)

The exact solution at time, tn, is given as

u(tn) = u0 cosω tn +u0

ωsinω tn. (4.2)

24

Accuracy of a numerical integration generally depends on the time step size, loading and the physical

parameters of the system [30].

4.1.2.1 Amplitude Error

Amplitude error can be measured by an equivalent viscous damping coefficient ξ, which is given by [1]

ξ = −ln(ρ (A))

ϕ(4.3)

where ϕ is the argument of the largest eigen value. ξ is a measure of the numerical damping ratio

introduced in the system through the integration scheme. Determination of amplitude decay can only

be done from the discrete solution of an initial-value problem. This necessitates post-processing which

involves approximate interpolation to ascertain consecutive peak values. Since ξ is defined in terms

of eigen values of the amplification matrix, hence it is the preferable measure for dissipation [55].

Reduction in amplitude is expressed as

δA = 1 − e−2πξ. (4.4)

For small step ratio ( hT ), reduction in amplitude can be defined as

δA ≈ 2πξ (4.5)

4.1.2.2 Period Error

The period error [1], introduced by the time integration scheme, is defined as,

δT

T=ω h

ϕ− 1. (4.6)

4.1.3 Damping

Damping dissipates energy causing the amplitude of free vibration to decay with time. Damping can

be inherent or deliberately added, perhaps to limit the peak response [30]. Damping the influences

structural dynamics can be categorized as follows.

1. Viscous Damping- Viscous damping exerts force proportional to velocity. Energy dissipated per

cycle is proportional to frequency and to the square of amplitude

2. Hysteresis/Solid Damping- Solid damping is inherent in the material and may result from plastic

action on a very small scale, with nominal stress in the elastic range. Energy dissipated per cycle

is independent of frequency.

3. Coulomb Damping- Coulomb damping resembles hysteresis damping but is associated with dry

damping.

25

4. Proportional Damping- The global damping matrix [C] is defined as a linear combination of the

global mass and stiffness matrices.

C = αM + βK (4.7)

This equation makes damping frequency dependent. The αM contribution damps the lowest

modes most heavily while the βK contribution damps the highest modes most heavily. Hence

βK term may be used to damp nonphysical high-frequency vibration from response simulations.

4.2 Stability and Accuracy Analysis of the Proposed Scheme

A single degree of freedom system with free vibration is considered to evaluate the stability of the

proposed scheme. The stability is evaluated by computing the eigenvalues of the amplification matrix

of the single degree of freedom system. The derivation of the amplification matrix for this case is

described next.

4.2.1 Amplification Matrix for the Proposed Scheme

Considering linear system the governing equation for the one degree of freedom system is expressed

as follows

M u + C u + K u = F . (4.8)

Considering tn+γt= tn + γt h as an instance of time between tn and tn+1 for γt ǫ (0, 1), Newmark scheme

is applied over the first substep γt h.. The equilibrium equation given by Eq. (4.8) is written at time

tn+γtas

M un+γt+ C un+γt

+ K un+γt= Fn+γt

. (4.9)

Substituting the equations for displacement and velocity of Newmark scheme, Eqs. (3.2) and (3.3), in

the equilibrium equation Eq. (4.9), we get,

[M + Cγ (γt h) + K β (γt h)2] un+γt+ [C + K (γt h)] un

+

[

C (γt h) (1 − γ) + K(γt h)2

2(1 − 2 β)

]

un + K un = Fn+γt. (4.10)

Here, β and γ are the Newmark parameters. From Eq. (4.10), the acceleration un+γtis obtained as

un+γt=

Fn+γt

[M + Cγ (γt h) + K β (γt h)2]−

K

[M + Cγ (γt h) + K β (γt h)2]un

−[C + K (γt h)]

[M + Cγ (γt h) + K β (γt h)2]un −

[

C (γt h) (1 − γ) + K(γt h)2

2 (1 − 2 β)

]

[M + Cγ (γt h) + K β (γt h)2]un , (4.11)

26

Substituting Eq. (4.11) in the approximation for displacement of Newmark scheme (Eq. 3.5) at tn+γt,

modified equation for displacement at tn+γtis obtained as

un+γt=

[

1 − β (γt h)2 K

[M + Cγ (γt h) + K β (γt h)2]

]

un

+

[

(γt h) − β (γt h)2 [C + K (γt h)]

[M + Cγ (γt h) + K β (γt h)2]

]

un

+

[ (γt h)2

2(1 − 2 β) − β (γt h)2

[

C (γt h) (1 − γ) + K(γt h)2

2 (1 − 2 β)

]

[M + Cγ (γt h) + K β (γt h)2]

]

un

+

[

β (γt h)2

[M + Cγ (γt h) + K β (γt h)2]

]

Fn+γt. (4.12)

Now, substituting Eq. (4.11) in the approximation for velocity, Eq. (3.4) at tn+γt, modified equation for

velocity at tn+γtis obtained as

un+γt= −

[

γ (γt h)K

[M + Cγ (γt h) + K β (γt h)2]

]

un

+

[

1 − γ (γt h)[C + K (γt h)]

[M + Cγ (γt h) + K β (γt h)2]

]

un

+

[

(1 − γ) (γt h) − γ (γt h)

[

C (γt h) (1 − γ) + K(γt h)2

2 (1 − 2 β)

]

[M + Cγ (γt h) + K β (γt h)2]

]

un

+

[

γ (γt h)

[M + Cγ (γt h) + K β (γt h)2]

]

Fn+γt(4.13)

In the second substep, the three point Backward Euler rule is applied over the second substep. The ap-

proximation for velocity and acceleration at time tn+ 1 is given by Eqs. (3.13) and (3.14). The equilibrium

equation given by Eq. (4.8) is now written at time tn+ 1 as

M un+ 1 + C un+ 1 + K un+ 1 = Fn+ 1. (4.14)

After substituting equations for velocity and acceleration of three point Backward Euler scheme in

Eq. (4.14) and after solving, the modified equation for displacement at tn+ 1 is obtained. Following

the same steps as in first substep for obtaining expressions for velocity and acceleration, equations

for velocity and acceleration at tn+ 1 is obtained. The equations at tn+ 1 are expressed in terms of

displacement, velocity and acceleration at tn. Note that for γt = 0.5, β = 0.25 and γ = 0.5, the

proposed scheme reduces to Bathe composite scheme [45, 46, 49].

27

Finally, the recursive relation between the quantities at time tn and tn+ 1 is obtained as

un+ 1

un+ 1

un+ 1

= A

un

un

un

,

where A is the amplification matrix and is expressed as following

A11 A12 A13

A21 A22 A23

A31 A32 A33

The characteristic equation of the amplification matrix is given by the following equation

|A − λ I| = 0 (4.15)

where λ and I are the eigenvalues of amplification matrix A and unit diagonal matrix, respectively.

For the proposed scheme, one of the terms A31 of amplification matrix is obtained as

A31 = −1

β1 β2

[

c2

(

2 (1 − γ)γt h + 4γξω (1 − γ) (γt h)2+ 2 β γ3

t ω2 h3 (1 − γ)

− 4 ξωγ (1 − γ)γ2t h2 − ω2 h3 γ3

t γ (1 − 2 β)

)

+ (c2 c3 + 2 ξω c2)

(

(γt h)2 (1 − 2 β) + 2 (γt h)2 (1 − 2 β) ξω hγγt

+ β (γt h)4ω2 (1 − 2 β) − 4 ξωβ (1 − γ) (γt h)3

− β (1 − 2 β) (γt h)4ω2

)]

(4.16)

where β1 = 1 + 2 ξω hγγt + βω2 h2 γ2t and β2 = c2

3+ 2 ξω c3 + ω2. Other terms of the amplification

matrix are obtained similar to A31.

The characteristic equation corresponding to Eq. (4.15) has eigenvalues λ1, λ2, and λ3 respectively.

The spectral radius ρ(A) should be less than unity for stability. Hence, a scheme is stable of the absolute

value of the eigenvalues are not greater than unity. The spectral radius of the proposed time integration

scheme is expressed as follows:

ρ(A) = max(||λ1|| , ||λ2|| , ||λ3||), (4.17)

where eigenvalues of matrix A are calculated from Eq. (4.15). The acceptable values of Newmark

parameters β and γ, and time step ratio γt for unconditional stability can be obtained from the stability

conditions (ρ(A) ≤ 1).

The stability and accuracy characteristics of the proposed scheme is now studied for various values

of the Newmark parameters β and γ as shown in Table 4.1. These values are chosen as per the following

28

Table 4.1: Newmark parameters.

S.No β γ1 0.25 0.52 0.3025 0.63 0.36 0.74 0.4225 0.85 0.49 0.9

relation [2]:

β ≥1

4

(

γ +1

2

)2

,

γ ≥1

2.

The values of γt considered for stability and accuracy analysis are chosen as 0.2, 0.4, 0.5, 0.6, and 0.8

respectively. The results for the proposed scheme are compared with those of the Bathe composite

scheme [46].

Figures 4.1(a), 4.2(a), 4.3(a), 4.4(a), and 4.5(a) show the plot of spectral radius with normalized

time step size for different values of γt for the proposed scheme. In Fig. 4.1(a), proposed scheme with

Newmark scheme paramters β = 0.25, γ = 0.5 and β = 0.3025, γ = 0.6 proves to be more efficient

in preserving the low frequency modes compared to Bathe composite scheme [46]. Also, it is observed

that as the value of Newmark parameters increases from β = 0.25, γ = 0.5 to β = 0.49, γ = 0.9,

the stability reduces in preserving the low frequency modes and damping the high frequency modes.

Proposed scheme with parameters γt = 0.8, β = 0.49 and γ = 0.9 (see Fig. 4.5(a)) is least efficient.

Figs. 4.1(b), 4.2(b), 4.3(b), 4.4(b), and 4.5(b) show the plot of amplitude error with normalized time

step size for different values of γt for different Newmark parameters and time step ratio (γt) values. In

Fig. 4.1(b), 4.2(b), and 4.5(b), for Newmark parameters β = 0.25, γ = 0.5 amplitude decay is quite

less than that of the Bathe composite scheme [46] . For all other Newmark parameters of the proposed

scheme (for all γt values) except for β = 0.25, γ = 0.5 , amplitude decays are high. For γt = 0.6,

β = 0.25 and γ = 0.5 (Fig. 4.4(b)), the amplitude decay of the proposed scheme is slightly more than

the Bathe composite scheme [46].

Figs. 4.1(c) to 4.5(c) show the period error with normalized time step size for different values of γt.

In Figs. 4.1(c) and 4.2(c), the period elongation for the proposed scheme is maximum for Newmark

parameters-β = 0.25 and γ = 0.5. For all other values of Newmark parameters, period elongation is

less than that of Bathe composite scheme.

For γt = 0.6 (Fig. 4.4(c)), period elongation of the proposed scheme is less than that of Bathe composite

scheme [46] for all values of Newmark parameters. For γt = 0.4 (Fig. 4.2(c)) and Newmark parameters-

β = 0.4225, γ = 0.8 and β = 0.49, γ = 0.9, period elongation is less than that of Bathe composite

scheme and for Newmark parameters-β = 0.36, γ = 0.7, period elongation almost coincides with the

29

0.01 0.1 1 10 100 1000

0.2

0.4

0.6

0.8

1

1.2

1.4

1.6

1.8

Time Ratio (∆ t/T)

Spe

ctra

l Rad

ius

ρ

Bathe schemeProposed scheme(γ

t = 0.2),(β −0.25),(γ −0.50)

Proposed scheme(γt = 0.2),(β −0.3025),(γ −0.6)




(a) Spectral radius

0 0.05 0.1 0.15 0.20

2

4

6

8

10

12

14

16

18

20


Am

plitu

de E

rror

(%)


t = 0.2),(β −0.25),(γ −0.5)





(b) Amplitude error

0 0.05 0.1 0.15 0.20

2

4

6

8

10

12

14

16

18

20


Per

iod

Err

or(%

)


t = 0.2),(β −0.25),(γ −0.5)





(c) Period error

Figure 4.1: Variation of spectral radii, amplitude error and period error for γt = 0.2.

Bathe composite scheme.

30

0.01 0.1 1 10 100 1000

0.2

0.4

0.6

0.8

1

1.2

1.4

1.6

1.8


Spe

ctra

l Rad

ius

ρ


t = 0.4),(β −0.25),(γ −0.50)





(a) Spectral radius

0 0.05 0.1 0.15 0.20

2

4

6

8

10

12

14

16

18

20


Am

plitu

de E

rror

(%)


t = 0.4),(β −0.25),(γ −0.5)





(b) Amplitude error

0 0.05 0.1 0.15 0.20

2

4

6

8

10

12

14

16

18

20


Per

iod

Err

or(%

)


t = 0.4),(β −0.25),(γ −0.5)





(c) Period error


31

0.01 0.1 1 10 100 1000

0.2

0.4

0.6

0.8

1

1.2

1.4

1.6

1.8


Spe

ctra

l Rad

ius

ρ


t = 0.5),(β −0.25),(γ −0.50)





(a) Spectral radius

0 0.05 0.1 0.15 0.20

2

4

6

8

10

12

14

16

18

20


Am

plitu

de E

rror

(%)


t = 0.5),(β −0.25),(γ −0.5)





(b) Amplitude error

0 0.05 0.1 0.15 0.20

2

4

6

8

10

12

14

16

18

20


Per

iod

Err

or(%

)


t = 0.5),(β −0.25),(γ −0.5)





(c) Period error


32

0.01 0.1 1 10 100 1000

0.2

0.4

0.6

0.8

1

1.2

1.4

1.6

1.8


Spe

ctra

l Rad

ius

ρ


t = 0.6),(β −0.25),(γ −0.50)





(a) Spectral radius

0 0.05 0.1 0.15 0.20

2

4

6

8

10

12

14

16

18

20


Am

plitu

de E

rror

(%)


t = 0.6),(β −0.25),(γ −0.5)





(b) Amplitude error

0 0.05 0.1 0.15 0.20

2

4

6

8

10

12

14

16

18

20


Per

iod

Err

or(%

)


t = 0.6),(β −0.25),(γ −0.5)





(c) Period error


33

0.01 0.1 1 10 100 1000

0.2

0.4

0.6

0.8

1

1.2

1.4

1.6

1.8


Spe

ctra

l Rad

ius

ρ


t = 0.8),(β −0.25),(γ −0.50)





(a) Spectral radius

0 0.05 0.1 0.15 0.20

2

4

6

8

10

12

14

16

18

20


Am

plitu

de E

rror

(%)


t = 0.8),(β −0.25),(γ −0.5)





(b) Amplitude error

0 0.05 0.1 0.15 0.20

2

4

6

8

10

12

14

16

18

20


Per

iod

Err

or(%

)


t = 0.8),(β −0.25),(γ −0.5)





(c) Period error


34

Chapter 5

Results and Discussion

In this chapter, the proposed scheme is implemented on two benchmark examples to discuss the overall

performance of the scheme. For this purpose, pendulum examples solved by Kuhl and Crisfield [26]

are discussed. These two examples - flexible pendulum and rigid pendulum - are classical geometrical

nonlinear problems to demonstrate the ability of any new time integration scheme to solve nonlin-

ear problems. The flexible pendulum problem is solved in Section 5.1 and by modifying the initial

conditions and the material stiffness, the second example, the ’stiff pendulum’ in Section 5.2, will be

obtained.

5.1 Numerical example: Flexible Pendulum

In this section, the proposed scheme is implemented to solve the flexible pendulum problem and

to examine the performance of the proposed scheme. The flexible pendulum problem is a classical

geometrical nonlinear example which involves large displacements and rotations. It is used to study

the ability of a time integration scheme for solving nonlinear problems. Formulations of this problem

has been analyzed by Kuhl and Crisfield [26].

The geometrical and physical characteristics of the elastic pendulum, the initial conditions, the bound-

ary conditions and other data are shown in Figure 5.1. For elastic pendulum, the stiffness E A is taken

as 104 N and the initial radial acceleration u0 as 0 m/s2 [26]. Elastic pendulum possess both high and

low frequency responses. For this two degree-of-freedom model, the first mode is represented by the

pendulum motion. The second mode, which contains high frequency responses, is represented by

axial motion [53]. Due to modified initial conditions, the pendulum will be loaded with centrifugal

force which induces high frequency vibration along the pendulum length. To capture the high axial

frequency, time steps considered are h = 0.0001, h = 0.01 seconds and h = 0.05 seconds. The substep

sizes taken are : γt = 0.2, 0.5 and 0.9. The transient analysis is done for a total time of 30 seconds.

Figures ( 5.2(a)- 5.2(b)), Figures ( 5.3(a)- 5.3(b)) and Figures ( 5.4(a)- 5.4(b)) show the variation of total

energy and angular momentum with time for h = 0.01s and γt = 0.2, 0.5 and 0.9 respectively. It is

35

Figure 5.1: Flexible pendulum. Data and initial conditions.

observed that for same time step, as the value of γt increases, numerical dissipation in total energy

and angular momentum also increases. This same behavior has been observed when h is changed to

0.05, see Figures ( 5.5(a)- 5.5(b)), Figures ( 5.6(a)- 5.6(b)) and Figures ( 5.7(a)- 5.7(b)). For very small

time step,i.e., h = 0.0001, numerical dissipation in total energy and angular momentum is very less,

see Figures ( 5.8(a)- 5.8(b)), Figures ( 5.9(a)- 5.9(b)) and Figures ( 5.10(a)- 5.10(b)). Also, as the value

of Newmark parameters increases, numerical dissipation increases. Maximum dissipation is for the

Newmark parameters (β,γ)=(0.49,0.9). No growth in energy and momentum of the system has been

observed. Hence, through higher numerical dissipation of the proposed scheme, better numerical

stability for the given non-linear problem can be obtained compared to Bathe composite scheme [46].

Figures (5.13(a) - 5.13(c)), Figures (5.11(a) - 5.11(c)) and Figures (5.12(a) - 5.12(c)) show the pendulum

trajectories for h = 0.0001, 0.01 and 0.05 respectively. It is observed that there is complete agreement of

the trajectories for the proposed scheme with that of the Bathe composite scheme [46].

Figures (5.14(a) - 5.14(c)) show the variation of axial strain of the pendulum with time for h = 0.0001

s and it depicts the errors in amplitude due to numerical dissipation. As the value of Newmark

parameters increases, the dissipation of axial strain with time increases. But for such a small time

step, the dissipation is minimal. For h = 0.1 s, as the value of Newmark parameters increases, the

36

0 5 10 15 20 25 300

50

100

150

200

250

300

Time(s)

Tot

al E

nerg

y(N

m)

β = 0.25 , γ = 0.5β = 0.3025 , γ = 0.6β = 0.36 , γ = 0.7β = 0.4225 , γ = 0.8β = 0.49 , γ = 0.9Bathe Scheme

(a) Total Energy

0 5 10 15 20 25 3080

100

120

140

160

180

200

220

240

Time(s)

Ang

ular

Mom

entu

m(N

ms)


(b) Angular Momentum

Figure 5.2: Variation of energy-momentum with time for h = 0.01 s and γt = 0.2.

0 5 10 15 20 25 300

50

100

150

200

250

300

Time(s)

Tot

al E

nerg

y(N

m)


(a) Total Energy

0 5 10 15 20 25 3080

100

120

140

160

180

200

220

240

Time(s)

Ang

ular

Mom

entu

m(N

ms)




variation of axial strain decreases. Least value of axial strain is observed for Newmark parameters(β,γ)

= (0.49,0.9).

5.2 Numerical example: Stiff Pendulum

In this section, the proposed scheme is implemented to solve the stiffpendulum problem and to examine

the performance of the proposed scheme. Formulations of this problem has been analyzed by Kuhl and

Crisfield [26]. The stiffness of the flexible pendulum (as shown in Fig. 5.1) is changed to E A = 1010N

and the initial acceleration, u0, is changed to 19.6 ms2 . Time steps considered are h = 0.0001 seconds and

h = 0.01 seconds. The substep sizes taken are : γt = 0.2, 0.5 and 0.9.

Figures ( 5.16(a)- 5.16(b)), Figures ( 5.17(a)- 5.17(b)) and Figures ( 5.18(a)- 5.18(b)) show the variation

of total energy and angular momentum with time for h=0.1s and γt=0.2, 0.5 and 0.9 respectively. It

is observed that for same time step, as the value of γt increases, numerical dissipation in total energy

37

0 5 10 15 20 25 300

50

100

150

200

250

300

Time(s)

Tot

al E

nerg

y(N

m)


(a) Total Energy

0 5 10 15 20 25 3080

100

120

140

160

180

200

220

240

Time(s)

Ang

ular

Mom

entu

m(N

ms)




0 5 10 15 20 25 300

50

100

150

200

250

300

Time(s)

Tot

al E

nerg

y(N

m)


(a) Total Energy

0 5 10 15 20 25 3080

100

120

140

160

180

200

220

240

Time(s)

Ang

ular

Mom

entu

m(N

ms)




and angular momentum also increases. This same behavior has been observed when h is changed

to 0.0001s, see Figures ( 5.19(a)- 5.19(b)), Figures ( 5.20(a)- 5.20(b)) and Figures ( 5.21(a)- 5.21(b)). But

the dissipation is minimal for total energy and angular momentum for h = 0.0001s Also, as the value

of Newmark parameters increases, numerical dissipation increases. Maximum dissipation is for the

Newmark parameters (β,γ)=(0.49,0.9). Hence, through higher numerical dissipation of the proposed

scheme, better numerical stability for the given non-linear problem can be obtained compared to Bathe

composite scheme [46].

Figures (5.22(a) - 5.22(c)) show the variation of axial strain of the pendulum with time for h =

0.0001s. Due to rigid-body motion, the magnitude of axial strain is very less. As the value of Newmark

parameters increases, the dissipation of axial strain with time increases. But for such a small time step,

the axial strain is minimal. Figures (5.23(a) - 5.23(c)) show the variation of axial strain of the pendulum

with time for h = 0.1s. For h = 0.1s, as the value of Newmark parameters increases, the variation

of axial strain is significant and clearly visible. Least value of axial strain is observed for Newmark

38

0 5 10 15 20 25 300

50

100

150

200

250

300

Time(s)

Tot

al E

nerg

y(N

m)


(a) Total Energy

0 5 10 15 20 25 3080

100

120

140

160

180

200

220

240

Time(s)

Ang

ular

Mom

entu

m(N

ms)




0 5 10 15 20 25 300

50

100

150

200

250

300

Time(s)

Tot

al E

nerg

y(N

m)


(a) Total Energy

0 5 10 15 20 25 3080

100

120

140

160

180

200

220

240

Time(s)

Ang

ular

Mom

entu

m(N

ms)


(b) Angular momentum


parameters(β,γ) = (0.49,0.9).

Figures (5.24(a) - 5.24(c)) and Figures (5.25(a) - 5.25(c)) show the pendulum trajectories for h =

0.0001s and 0.1s respectively. It is observed that there is complete agreement of the trajectories for

the proposed scheme for h = 0.0001s with that of the Bathe composite scheme [46] but for h = 0.1s,

there is slight variation in trajectories compared to Bathe composite scheme, which may be a result of

inaccuracy for larger time step.

39

0 5 10 15 20 25 300

50

100

150

200

250

300

Time(s)

Tot

al E

nerg

y(N

m)


(a) Total Energy

0 5 10 15 20 25 3080

100

120

140

160

180

200

220

240

Time(s)

Ang

ular

Mom

entu

m(N

ms)




0 5 10 15 20 25 300

50

100

150

200

250

300

Time(s)

Tot

al E

nerg

y(N

m)


(a) Total Energy

0 5 10 15 20 25 3080

100

120

140

160

180

200

220

240

Time(s)

Ang

ular

Mom

entu

m(N

ms)




0 5 10 15 20 25 300

50

100

150

200

250

300

Time(s)

Tot

al E

nerg

y(N

m)


(a) Total Energy

0 5 10 15 20 25 3080

100

120

140

160

180

200

220

240

Time(s)

Ang

ular

Mom

entu

m(N

ms)


(b) Angular momentum


40

−4 −3 −2 −1 0 1 2 3 4−4

−3

−2

−1

0

1

2

3

4

coordinate x (m)

coor

dina

te y

(m

)


(a) γt = 0.2

−4 −3 −2 −1 0 1 2 3 4−4

−3

−2

−1

0

1

2

3

4

coordinate x (m)

coor

dina

te y

(m

)


(b) γt = 0.5

−4 −3 −2 −1 0 1 2 3 4−4

−3

−2

−1

0

1

2

3

4

coordinate x (m)

coor

dina

te y

(m

)


(c) γt = 0.9

Figure 5.11: Variation of trajectory of the pendulum for h = 0.01 s.

41

−4 −3 −2 −1 0 1 2 3 4−4

−3

−2

−1

0

1

2

3

4

coordinate x (m)

coor

dina

te y

(m

)


(a) γt = 0.2

−4 −3 −2 −1 0 1 2 3 4−4

−3

−2

−1

0

1

2

3

4

coordinate x (m)

coor

dina

te y

(m

)


(b) γt = 0.5

−4 −3 −2 −1 0 1 2 3 4−4

−3

−2

−1

0

1

2

3

4

coordinate x (m)

coor

dina

te y

(m

)


(c) γt = 0.9


42

−4 −3 −2 −1 0 1 2 3 4−4

−3

−2

−1

0

1

2

3

4

coordinate x (m)

coor

dina

te y

(m

)


(a) γt = 0.2

−4 −3 −2 −1 0 1 2 3 4−4

−3

−2

−1

0

1

2

3

4

coordinate x (m)

coor

dina

te y

(m

)


(b) γt = 0.5

−4 −3 −2 −1 0 1 2 3 4−4

−3

−2

−1

0

1

2

3

4

coordinate x (m)

coor

dina

te y

(m

)


(c) γt = 0.9


43

0 2.5 5 7.5 10 12.5 150

0.005

0.01

0.015

0.02

0.025

0.03

0.035

0.04

0.045

0.05

Time(s)

Str

ain

ε


(a) γt = 0.2

0 2.5 5 7.5 10 12.5 150

0.005

0.01

0.015

0.02

0.025

0.03

0.035

0.04

0.045

0.05

Time(s)

Str

ain

ε


(b) γt = 0.5

0 2.5 5 7.5 10 12.5 150

0.005

0.01

0.015

0.02

0.025

0.03

0.035

0.04

0.045

0.05

Time(s)

Str

ain

ε


(c) γt = 0.9

Figure 5.14: Variation of strain with time for h = 0.0001 s.

44

0 2.5 5 7.5 10 12.5 150

0.005

0.01

0.015

0.02

0.025

0.03

0.035

0.04

0.045

0.05

Time(s)

Str

ain

ε


(a) γt = 0.2

0 2.5 5 7.5 10 12.5 150

0.005

0.01

0.015

0.02

0.025

0.03

0.035

0.04

0.045

0.05

Time(s)

Str

ain

ε


(b) γt = 0.5

0 2.5 5 7.5 10 12.5 150

0.005

0.01

0.015

0.02

0.025

0.03

0.035

0.04

0.045

0.05

Time(s)

Str

ain

ε


(c) γt = 0.9

Figure 5.15: Variation of strain with time for h = 0.01 s.

0 5 10 15 20 25 30 35 40 45 500

50

100

150

200

250

300

Time(s)

Tot

al E

nerg

y(N

m)


(a) Total Energy

0 5 10 15 20 25 30 35 40 45 500

20

40

60

80

100

120

140

160

180

200

220

240

Time(s)

Ang

ular

Mom

entu

m(N

ms)




45

0 5 10 15 20 25 30 35 40 45 500

50

100

150

200

250

300

Time(s)

Tot

al E

nerg

y(N

m)


(a) Total Energy

0 5 10 15 20 25 30 35 40 45 500

20

40

60

80

100

120

140

160

180

200

220

240

Time(s)

Ang

ular

Mom

entu

m(N

ms)




0 5 10 15 20 25 30 35 40 45 500

50

100

150

200

250

300

Time(s)

Tot

al E

nerg

y(N

m)


(a) Total Energy

0 5 10 15 20 25 30 35 40 45 500

20

40

60

80

100

120

140

160

180

200

220

240

Time(s)

Ang

ular

Mom

entu

m(N

ms)




0 5 10 15 20 25 300

50

100

150

200

250

300

Time(s)

Tot

al E

nerg

y(N

m)


(a) Total Energy

0 5 10 15 20 25 300

20

40

60

80

100

120

140

160

180

200

220

240

Time(s)

Ang

ular

Mom

entu

m(N

ms)




46

0 5 10 15 20 25 300

50

100

150

200

250

300

Time(s)

Tot

al E

nerg

y(N

m)


(a) Total Energy

0 5 10 15 20 25 300

20

40

60

80

100

120

140

160

180

200

220

240

Time(s)

Ang

ular

Mom

entu

m(N

ms)




0 5 10 15 20 25 300

50

100

150

200

250

300

Time(s)

Tot

al E

nerg

y(N

m)


(a) Total Energy

0 5 10 15 20 25 300

20

40

60

80

100

120

140

160

180

200

220

240

Time(s)

Ang

ular

Mom

entu

m(N

ms)




47

0 5 10 15 20 25 30−2e−08

−1e−08

0

1e−08

2e−08

3e−08

Time(s)

Str

ain

ε


(a) γt = 0.2

0 5 10 15 20 25 30−2e−08

−1e−08

0

1e−08

2e−08

3e−08

Time(s)

Str

ain

ε


(b) γt = 0.5

0 5 10 15 20 25 30−2e−08

−1e−08

0

1e−08

2e−08

3e−08

Time(s)

Str

ain

ε


(c) γt = 0.9

Figure 5.22: Variation of axial strain with time for h = 0.0001 s.

48

0 5 10 15 20 25 30 35 40 45 50−2e−08

−1e−08

0

1e−08

2e−08

3e−08

Time(s)

Str

ain

ε


(a) γt = 0.2

0 5 10 15 20 25 30 35 40 45 50−2e−08

−1e−08

0

1e−08

2e−08

3e−08

Time(s)

Str

ain

ε


(b) γt = 0.5

0 5 10 15 20 25 30 35 40 45 50−2e−08

−1e−08

0

1e−08

2e−08

3e−08

Time(s)

Str

ain

ε


(c) γt = 0.9

Figure 5.23: Variation of axial strain with time for h = 0.1 s.

49

−4 −3 −2 −1 0 1 2 3 4−4

−3

−2

−1

0

1

2

3

4

coordinate x (m)

coor

dina

te y

(m

)


(a) γt = 0.2

−4 −3 −2 −1 0 1 2 3 4−4

−3

−2

−1

0

1

2

3

4

coordinate x (m)

coor

dina

te y

(m

)


(b) γt = 0.5

−4 −3 −2 −1 0 1 2 3 4−4

−3

−2

−1

0

1

2

3

4

coordinate x (m)

coor

dina

te y

(m

)


(c) γt = 0.9


50

−4 −3 −2 −1 0 1 2 3 4−4

−3

−2

−1

0

1

2

3

4

coordinate x (m)

coor

dina

te y

(m

)


(a) γt = 0.2

−4 −3 −2 −1 0 1 2 3 4−4

−3

−2

−1

0

1

2

3

4

coordinate x (m)

coor

dina

te y

(m

)


(b) γt = 0.5

−4 −3 −2 −1 0 1 2 3 4−4

−3

−2

−1

0

1

2

3

4

coordinate x (m)

coor

dina

te y

(m

)


(c) γt = 0.9


51

Chapter 6

Conclusions and Scope for the Future

Work

6.1 Summary

In the present work, a new composite time integration scheme has been proposed. The characteristics

of the proposed scheme i.e., stability and accuracy are studied and compared with Bathe composite

scheme [46]. Different combinations of Newmark parameters and substep sizes for the proposed

scheme have been studied. This scheme gives freedom to choose any combinations of Newmark

parameters and substep sizes to control the high frequency dissipation. For some combinations of

Newmark parameters and substep sizes, the proposed scheme gives better results in terms of stability

and accuracy when compared with Bathe composite scheme [46]. The proposed scheme is applied to

two nonlinear dynamic problems i.e., flexible pendulum and stiff pendulum. This scheme gives more

flexibility to vary the dissipation aspect by choosing different combinations of Newmark parameters

and γt values. The proposed scheme gives more numerical stability compared to Bathe composite

scheme [46]. The performance of the scheme is studied for different values of γt on energy and

momentum conservation. For a particular time step, as the value of γt increases, numerical dissipation

also increases. Numerical dissipation also increases with the increase of Newmark parameters. It can

also be concluded from the present study that use of too large time step leads to excessive numerical

dissipation.

6.2 Scope of the Future Work

Further work can be carried out on implementation of the proposed scheme to contact-impact problems.

Also, more studies can be carried out to assess the energy and momentum conservation aspects.

Comparative studies of the proposed scheme with some more time integration schemes can also be

carried out. Error estimators for the proposed scheme can be developed. This will be helpful in design

of an adaptive time-stepping strategy based on current scheme.

52

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56

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