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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
4.2 Variation of spectral radii, amplitude error and period error for γt = 0.4. . . . . . . . . . 31
4.3 Variation of spectral radii, amplitude error and period error for γt = 0.5. . . . . . . . . . 32
4.4 Variation of spectral radii, amplitude error and period error for γt = 0.6. . . . . . . . . . 33
4.5 Variation of spectral radii, amplitude error and period error for γt = 0.8. . . . . . . . . . 34
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
5.3 Variation of energy-momentum with time for h = 0.01 s and γt = 0.5. . . . . . . . . . . 37
5.4 Variation of energy-momentum with time for h = 0.01 s and γt = 0.9. . . . . . . . . . . 38
5.5 Variation of energy-momentum with time for h = 0.05 s and γt = 0.2. . . . . . . . . . . 38
5.6 Variation of energy-momentum with time for h = 0.05 s and γt = 0.5. . . . . . . . . . . 39
5.7 Variation of energy-momentum with time for h = 0.05 s and γt = 0.9. . . . . . . . . . . 39
5.8 Variation of energy-momentum with time for h = 0.0001 s and γt = 0.2. . . . . . . . . . 40
5.9 Variation of energy-momentum with time for h = 0.0001 s and γt = 0.5. . . . . . . . . . 40
5.10 Variation of energy-momentum with time for h = 0.0001 s and γt = 0.9. . . . . . . . . . 40
5.11 Variation of trajectory of the pendulum for h = 0.01 s. . . . . . . . . . . . . . . . . . . . . 41
5.12 Variation of trajectory of the pendulum for h = 0.05 s. . . . . . . . . . . . . . . . . . . . . 42
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
5.17 Variation of energy-momentum with time for h = 0.1 s and γt = 0.5. . . . . . . . . . . . 46
5.18 Variation of energy-momentum with time for h = 0.1 s and γt = 0.9. . . . . . . . . . . . 46
xiii
5.19 Variation of energy-momentum with time for h = 0.0001 s and γt = 0.2. . . . . . . . . . 46
5.20 Variation of energy-momentum with time for h = 0.0001 s and γt = 0.5. . . . . . . . . . 47
5.21 Variation of energy-momentum with time for h = 0.0001 s and γt = 0.9. . . . . . . . . . 47
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
5.25 Variation of trajectory of the pendulum for h = 0.1 s. . . . . . . . . . . . . . . . . . . . . 51
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)
Proposed scheme(γt = 0.2),(β −0.36),(γ −0.7)
Proposed scheme(γt = 0.2),(β −0.4225),(γ −0.8)
Proposed scheme(γt = 0.2),(β −0.49),(γ −0.9)
(a) Spectral radius
0 0.05 0.1 0.15 0.20
2
4
6
8
10
12
14
16
18
20
Time Ratio (∆ t/T)
Am
plitu
de E
rror
(%)
Bathe schemeProposed scheme(γ
t = 0.2),(β −0.25),(γ −0.5)
Proposed scheme(γt = 0.2),(β −0.3025),(γ −0.6)
Proposed scheme(γt = 0.2),(β −0.36),(γ −0.7)
Proposed scheme(γt = 0.2),(β −0.4225),(γ −0.8)
Proposed scheme(γt = 0.2),(β −0.49),(γ −0.9)
(b) Amplitude error
0 0.05 0.1 0.15 0.20
2
4
6
8
10
12
14
16
18
20
Time Ratio (∆ t/T)
Per
iod
Err
or(%
)
Bathe schemeProposed scheme(γ
t = 0.2),(β −0.25),(γ −0.5)
Proposed scheme(γt = 0.2),(β −0.3025),(γ −0.6)
Proposed scheme(γt = 0.2),(β −0.36),(γ −0.7)
Proposed scheme(γt = 0.2),(β −0.4225),(γ −0.8)
Proposed scheme(γt = 0.2),(β −0.49),(γ −0.9)
(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
Time Ratio (∆ t/T)
Spe
ctra
l Rad
ius
ρ
Bathe schemeProposed scheme(γ
t = 0.4),(β −0.25),(γ −0.50)
Proposed scheme(γt = 0.4),(β −0.3025),(γ −0.6)
Proposed scheme(γt = 0.4),(β −0.36),(γ −0.7)
Proposed scheme(γt = 0.4),(β −0.4225),(γ −0.8)
Proposed scheme(γt = 0.4),(β −0.49),(γ −0.9)
(a) Spectral radius
0 0.05 0.1 0.15 0.20
2
4
6
8
10
12
14
16
18
20
Time Ratio (∆ t/T)
Am
plitu
de E
rror
(%)
Bathe schemeProposed scheme(γ
t = 0.4),(β −0.25),(γ −0.5)
Proposed scheme(γt = 0.4),(β −0.3025),(γ −0.6)
Proposed scheme(γt = 0.4),(β −0.36),(γ −0.7)
Proposed scheme(γt = 0.4),(β −0.4225),(γ −0.8)
Proposed scheme(γt = 0.4),(β −0.49),(γ −0.9)
(b) Amplitude error
0 0.05 0.1 0.15 0.20
2
4
6
8
10
12
14
16
18
20
Time Ratio (∆ t/T)
Per
iod
Err
or(%
)
Bathe schemeProposed scheme(γ
t = 0.4),(β −0.25),(γ −0.5)
Proposed scheme(γt = 0.4),(β −0.3025),(γ −0.6)
Proposed scheme(γt = 0.4),(β −0.36),(γ −0.7)
Proposed scheme(γt = 0.4),(β −0.4225),(γ −0.8)
Proposed scheme(γt = 0.4),(β −0.49),(γ −0.9)
(c) Period error
Figure 4.2: Variation of spectral radii, amplitude error and period error for γt = 0.4.
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
Time Ratio (∆ t/T)
Spe
ctra
l Rad
ius
ρ
Bathe schemeProposed scheme(γ
t = 0.5),(β −0.25),(γ −0.50)
Proposed scheme(γt = 0.5),(β −0.3025),(γ −0.6)
Proposed scheme(γt = 0.5),(β −0.36),(γ −0.7)
Proposed scheme(γt = 0.5),(β −0.4225),(γ −0.8)
Proposed scheme(γt = 0.5),(β −0.49),(γ −0.9)
(a) Spectral radius
0 0.05 0.1 0.15 0.20
2
4
6
8
10
12
14
16
18
20
Time Ratio (∆ t/T)
Am
plitu
de E
rror
(%)
Bathe schemeProposed scheme(γ
t = 0.5),(β −0.25),(γ −0.5)
Proposed scheme(γt = 0.5),(β −0.3025),(γ −0.6)
Proposed scheme(γt = 0.5),(β −0.36),(γ −0.7)
Proposed scheme(γt = 0.5),(β −0.4225),(γ −0.8)
Proposed scheme(γt = 0.5),(β −0.49),(γ −0.9)
(b) Amplitude error
0 0.05 0.1 0.15 0.20
2
4
6
8
10
12
14
16
18
20
Time Ratio (∆ t/T)
Per
iod
Err
or(%
)
Bathe schemeProposed scheme(γ
t = 0.5),(β −0.25),(γ −0.5)
Proposed scheme(γt = 0.5),(β −0.3025),(γ −0.6)
Proposed scheme(γt = 0.5),(β −0.36),(γ −0.7)
Proposed scheme(γt = 0.5),(β −0.4225),(γ −0.8)
Proposed scheme(γt = 0.5),(β −0.49),(γ −0.9)
(c) Period error
Figure 4.3: Variation of spectral radii, amplitude error and period error for γt = 0.5.
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
Time Ratio (∆ t/T)
Spe
ctra
l Rad
ius
ρ
Bathe schemeProposed scheme(γ
t = 0.6),(β −0.25),(γ −0.50)
Proposed scheme(γt = 0.6),(β −0.3025),(γ −0.6)
Proposed scheme(γt = 0.6),(β −0.36),(γ −0.7)
Proposed scheme(γt = 0.6),(β −0.4225),(γ −0.8)
Proposed scheme(γt = 0.6),(β −0.49),(γ −0.9)
(a) Spectral radius
0 0.05 0.1 0.15 0.20
2
4
6
8
10
12
14
16
18
20
Time Ratio (∆ t/T)
Am
plitu
de E
rror
(%)
Bathe schemeProposed scheme(γ
t = 0.6),(β −0.25),(γ −0.5)
Proposed scheme(γt = 0.6),(β −0.3025),(γ −0.6)
Proposed scheme(γt = 0.6),(β −0.36),(γ −0.7)
Proposed scheme(γt = 0.6),(β −0.4225),(γ −0.8)
Proposed scheme(γt = 0.6),(β −0.49),(γ −0.9)
(b) Amplitude error
0 0.05 0.1 0.15 0.20
2
4
6
8
10
12
14
16
18
20
Time Ratio (∆ t/T)
Per
iod
Err
or(%
)
Bathe schemeProposed scheme(γ
t = 0.6),(β −0.25),(γ −0.5)
Proposed scheme(γt = 0.6),(β −0.3025),(γ −0.6)
Proposed scheme(γt = 0.6),(β −0.36),(γ −0.7)
Proposed scheme(γt = 0.6),(β −0.4225),(γ −0.8)
Proposed scheme(γt = 0.6),(β −0.49),(γ −0.9)
(c) Period error
Figure 4.4: Variation of spectral radii, amplitude error and period error for γt = 0.6.
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
Time Ratio (∆ t/T)
Spe
ctra
l Rad
ius
ρ
Bathe schemeProposed scheme(γ
t = 0.8),(β −0.25),(γ −0.50)
Proposed scheme(γt = 0.8),(β −0.3025),(γ −0.6)
Proposed scheme(γt = 0.8),(β −0.36),(γ −0.7)
Proposed scheme(γt = 0.8),(β −0.4225),(γ −0.8)
Proposed scheme(γt = 0.8),(β −0.49),(γ −0.9)
(a) Spectral radius
0 0.05 0.1 0.15 0.20
2
4
6
8
10
12
14
16
18
20
Time Ratio (∆ t/T)
Am
plitu
de E
rror
(%)
Bathe schemeProposed scheme(γ
t = 0.8),(β −0.25),(γ −0.5)
Proposed scheme(γt = 0.8),(β −0.3025),(γ −0.6)
Proposed scheme(γt = 0.8),(β −0.36),(γ −0.7)
Proposed scheme(γt = 0.8),(β −0.4225),(γ −0.8)
Proposed scheme(γt = 0.8),(β −0.49),(γ −0.9)
(b) Amplitude error
0 0.05 0.1 0.15 0.20
2
4
6
8
10
12
14
16
18
20
Time Ratio (∆ t/T)
Per
iod
Err
or(%
)
Bathe schemeProposed scheme(γ
t = 0.8),(β −0.25),(γ −0.5)
Proposed scheme(γt = 0.8),(β −0.3025),(γ −0.6)
Proposed scheme(γt = 0.8),(β −0.36),(γ −0.7)
Proposed scheme(γt = 0.8),(β −0.4225),(γ −0.8)
Proposed scheme(γt = 0.8),(β −0.49),(γ −0.9)
(c) Period error
Figure 4.5: Variation of spectral radii, amplitude error and period error for γt = 0.8.
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)
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
(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)
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)
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
(b) Angular Momentum
Figure 5.3: Variation of energy-momentum with time for h = 0.01 s and γt = 0.5.
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)
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)
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
(b) Angular Momentum
Figure 5.4: Variation of energy-momentum with time for h = 0.01 s and γt = 0.9.
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)
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
(b) Angular Momentum
Figure 5.5: Variation of energy-momentum with time for h = 0.05 s and γt = 0.2.
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)
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)
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
(b) Angular Momentum
Figure 5.6: Variation of energy-momentum with time for h = 0.05 s and γt = 0.5.
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)
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
(b) Angular momentum
Figure 5.7: Variation of energy-momentum with time for h = 0.05 s and γt = 0.9.
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)
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)
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
(b) Angular Momentum
Figure 5.8: Variation of energy-momentum with time for h = 0.0001 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)
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)
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
(b) Angular Momentum
Figure 5.9: Variation of energy-momentum with time for h = 0.0001 s and γt = 0.5.
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)
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
(b) Angular momentum
Figure 5.10: Variation of energy-momentum with time for h = 0.0001 s and γt = 0.9.
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
)
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) 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
)
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
(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
)
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
(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
)
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) 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
)
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
(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
)
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
(c) γt = 0.9
Figure 5.12: Variation of trajectory of the pendulum for h = 0.05 s.
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
)
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) 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
)
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
(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
)
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
(c) γt = 0.9
Figure 5.13: Variation of trajectory of the pendulum for h = 0.0001 s.
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
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) 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
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
(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
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
(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
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) 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
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
(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
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
(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)
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 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)
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
(b) Angular Momentum
Figure 5.16: Variation of energy-momentum with time for h = 0.1 s and γt = 0.2.
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)
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 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)
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
(b) Angular Momentum
Figure 5.17: Variation of energy-momentum with time for h = 0.1 s and γt = 0.5.
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)
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 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)
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
(b) Angular Momentum
Figure 5.18: Variation of energy-momentum with time for h = 0.1 s and γt = 0.9.
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 25 300
20
40
60
80
100
120
140
160
180
200
220
240
Time(s)
Ang
ular
Mom
entu
m(N
ms)
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
(b) Angular Momentum
Figure 5.19: Variation of energy-momentum with time for h = 0.0001 s and γt = 0.2.
46
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 25 300
20
40
60
80
100
120
140
160
180
200
220
240
Time(s)
Ang
ular
Mom
entu
m(N
ms)
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
(b) Angular Momentum
Figure 5.20: Variation of energy-momentum with time for h = 0.0001 s and γt = 0.5.
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 25 300
20
40
60
80
100
120
140
160
180
200
220
240
Time(s)
Ang
ular
Mom
entu
m(N
ms)
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
(b) Angular Momentum
Figure 5.21: Variation of energy-momentum with time for h = 0.0001 s and γt = 0.9.
47
0 5 10 15 20 25 30−2e−08
−1e−08
0
1e−08
2e−08
3e−08
Time(s)
Str
ain
ε
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) 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
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
(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
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
(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
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) 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
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
(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
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
(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
)
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) 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
)
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
(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
)
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
(c) γt = 0.9
Figure 5.24: Variation of trajectory of the pendulum for h = 0.0001 s.
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
)
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) 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
)
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
(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
)
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
(c) γt = 0.9
Figure 5.25: Variation of trajectory of the pendulum for h = 0.1 s.
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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