A Time Integration Scheme for Dynamic Problems

α Parameter for Chung and Hulbert Generalised-α schemeαg Parameter for Gohlampour composite scheme β Parameter for Newmark scheme γ Parameter for Newmark scheme γt Time step ratio for th

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

Dedicated to

My Parents

and to

Dr Pankaj Biswas and Dr Rashmi Ranjan Das

Teachers and Friends

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 therest of my life

I am specially grateful to Prof Pankaj Biswas for his support, encouragement and inspiring adviceswhich 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 forimparting me knowledge of various subjects and helping me at the time of difficulty in solving anyproblem 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 andsupport

I am thankful to my parents, Shri A Mohan Rao and Smt A Sarita, for providing me support andencouragement at every step of my life I am also thankful to my seniors, Dipendra Kumar Roy, VinayMishra, Sibananda Mohanty, Manish Kumar Dubey, Sunil Kumar Singh, Debabrata Gayen, SusantaBehera 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, AnuragMishra, Nikhil Sharma, Abhishek Yadav, Sateesh Kumar, Vivek Badhe and Susobhan Patra Finally, Iexpress my thanks to all those who have helped me directly or indirectly for successful completion ofthis work

Sandeep Kumar IIT Guwahati May, 2015

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).

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

6 Conclusions and Scope for the Future Work 52

6.1 Summary 526.2 Scope of the Future Work 52

List of Figures

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

4.1 Variation of spectral radii, amplitude error and period error forγt = 0.2 30

4.2 Variation of spectral radii, amplitude error and period error forγt = 0.4 31

4.3 Variation of spectral radii, amplitude error and period error forγt = 0.5 32

4.4 Variation of spectral radii, amplitude error and period error forγt = 0.6 33

4.5 Variation of spectral radii, amplitude error and period error forγt= 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 γ t = 0.2 37

5.3 Variation of energy-momentum with time for h = 0.01 s and γ t = 0.5 37

5.4 Variation of energy-momentum with time for h = 0.01 s and γ t = 0.9 38

5.5 Variation of energy-momentum with time for h = 0.05 s and γ t = 0.2 38

5.6 Variation of energy-momentum with time for h = 0.05 s and γ t = 0.5 39

5.7 Variation of energy-momentum with time for h = 0.05 s and γ t = 0.9 39

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

5.9 Variation of energy-momentum with time for h = 0.0001 s and γ t = 0.5 40

5.10 Variation of energy-momentum with time for h = 0.0001 s and γ t = 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 γ t = 0.2 45

5.17 Variation of energy-momentum with time for h = 0.1 s and γ t = 0.5 46

5.18 Variation of energy-momentum with time for h = 0.1 s and γ t = 0.9 46

5.19 Variation of energy-momentum with time for h = 0.0001 s and γ t = 0.2 46

5.20 Variation of energy-momentum with time for h = 0.0001 s and γ t = 0.5 47

5.21 Variation of energy-momentum with time for h = 0.0001 s and γ t = 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

List of Tables

4.1 Newmark parameters 29

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

ξ Modal damping ratio

ω Vibration frequency

Chapter 1

Introduction

Transient response analysis is used to compute the dynamic response of a structure subjected to varying excitation The distinctive nature between static and dynamic problem is the presence of inertiaforces in dynamic problem which opposes the motion generated by the applied dynamic loading Thedynamic nature of a problem is dominant if the inertia forces are large compared to the total appliedforces When the motion generated by the applied forces are small such that the inertia forces arenegligible then the problem is considered as static [1]

time-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 timeintegration schemes are discussed Then, a detailed classification of direct time integration schemes ispresented in 2.1 Both the explicit and implicit schemes are discussed Section 2.2.2 discusses variousimplicit time integration schemes which are the focus of this research The objective of the work arediscussed 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 ofthe 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,

only the lower frequencies and modes of the structure need to be retained Modes retained musthave frequencies that span the temporal variation of loading Mode shapes of free vibration are notrelated to the complexity of the loading The number of modes should be enough to approximate thedisplacement associated with spatial variation of loading.

In modal superposition method, a set of uncoupled equations are obtained from the coupledequations of motion of a discrete degree of freedom system using a transformation into modal ornormal coordinate space In this space, each mode responds to its own mode shape φi, vibrationfrequencyωi, and modal damping ξi The total response can be obtained by summation of all the singledegree 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 isapplicable for linear elastic systems

For linear dynamic problems, where the response is dominated by low frequencies, mode position method can be used to reduce the computational cost without sacrificing the accuracy Butapplication 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 timeintegration schemes prove to be more reliable and efficient

super-1.1.2 Direct Time Integration

Modal methods use reduced set of degree of freedom to determine the displacements, velocities, andaccelerations as a function of time and then transform them back into original physical degree offreedom space On the hand in direct time integration scheme, no such transformation of equation ofmotion is carried out The response history is calculated using step-by-step integration in time Theneed 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 alsoused to directly calculate the response of systems with large number of degree of freedoms to avoidtime-consuming calculations of eigenvalues and eigenvectors of the systems Moreover, there is noneed to compute modes and frequencies in time integration scheme

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

where u is the displacement vector, M, C, and K are the mass, damping and stiffness matrices tively, and F is the vector of externally applied loads The superimposed dots denotes derivative with

respec-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

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 isproposed to combine the Newmark scheme [4] with the three point backward Euler scheme tohave more user controlled numerical properties like high frequency dissipation

2 To study the stability, accuracy, and dissipation of the proposed scheme This is achieved bystudying 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 integrationschemes 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 ispresentee Various schemes in each class are detailed Finally, a detailed review of some recent implicittime 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 isdiscussed 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

as in for e.g., automobile crash simulation, design of landing gears for airplanes and space crafts, tirewear 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 offreedom systems [1] They determine the approximate values of the exact solution at discrete timeintervals 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 [t n , t n+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

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

the equation of motion is developed over the time interval [t n , t n+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 timeaverages [5] The monograph by Wood [6] gives a detailed survey and mathematical background ofvarious implicit and explicit schemes developed until 1990 A third family of methods i.e., Galerkinmethods 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 isstill extensively used [4] It is known that even for the linear case, the Newmark scheme, or thosebased on it, the energy is only conserved for a particular choice of the Newmark parametersβ = 1

4andγ = 1

2 Even for this choice of parameters energy is not conserved for nonlinear systems In manyapplications only lower mode response is of interest In such cases temporal integration schemes havebeen developed with a controllable numerical dissipation for higher modes (see for e.g., [10–15]) Taking

γ > 12 andβ > 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 conservationand dissipation in linear Newmark-type algorithms and theirα modifications is discussed in [16]

It is well known that the traditional temporal discretization schemes like the Newmark basedschemes, which are unconditionally stable for linear problems, exhibit significant instabilities whenapplied to nonlinear elastodynamics problems [17–19] This has led to a significant amount of researchover the past two decades to develop more robust temporal discretization schemes for nonlinearelastodynamic 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 wayinto 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, bydesign, the conservation laws of momentum along with an a-priori estimate on the rate of decay ofthe total energy They called these algorithms energy momentum conserving algorithms (EMCA) Theproposed methodology considered a Saint Venant-Kirchhoff elasticity model This scheme was furtherextended to general elastic materials [22], systems with constraints [23, 24], shells [25, 26], compositelaminates [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.

2.2.1 Explicit Time Integration Schemes

General form of difference equation for explicit scheme is expressed as

un+1= f (u n, ˙un, ¨un, un−1, ˙un−1, ) (2.1)

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

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

by substituting these values in Eq 1.1 and solving system of simultaneous linear equations In explicitscheme, solver (direct or iterative) is not needed since the mass matrix is diagonalized and the variablescan be found simply dividing with force vector In general, most explicit schemes are conditionallystable 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 sizeunder 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 tradeoff is done with numerical accuracy Explicit schemes [30] are suitable for wave propagation problemswhere all modes participate in the solution Some of the examples of explicit schemes are Centraldifference scheme, Forward Euler scheme, Runge-Kutta scheme etc The details of these schemes arepresented next

h23

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

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

at the previous step time t = t n−1 Approximation to the solution un+1 at the time t = t n+1isgiven as

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

The second-order Runge-Kutta scheme has third-order local truncation error O(h3) The order Runge-Kutta scheme is one-step explicit scheme [29] and is given as

2 , t n +1

,

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

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

at time t nis given as

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

the solution un+1is given follows

2.2.2 Implicit Time Integration Schemes

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

un+1 = f (u n, ˙un, ¨un, un−1, un+1, ) (2.12)

In the implicit schemes, the displacements and the velocities at the current time step are expressednot 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-Rapshonmethod, 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

method Implicit schemes are suitable for structural dynamics problems (inertial or vibrations typeapplications) 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 forimplicit 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 stillextensively 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 parametersβ = 1

4 and

γ = 1

2 Even for this choice of parameters energy is not conserved for nonlinear systems In manyapplications only lower mode response is of interest In such cases temporal integration schemes havebeen developed with a controllable numerical dissipation for higher modes (see for e.g., [10–15]) Taking

γ > 12 andβ > 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 conservationand dissipation in linear Newmark-type algorithms and theirα modifications is discussed in [16]

A comprehensive study on direct time integration schemes have been done by Subbaraj and ish [31] and Bert [32] They have done comparative evaluation of different time integration schemesalong with their implementation to some numerical problems Another important characteristic of timeintegration schemes i.e., overshooting, have been studied by Hilber [33] Also, along with overshootingcharacteristic, 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 matrixnumerically and discussed the overshooting effects This is a powerful method to check whether thealgorithm 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 schemepopularly known as HHT-α scheme This scheme is unconditionally stable and it has been developedfor better preservation of low frequency modes Another scheme which is a modification of Newmarkscheme has been developed by Wood et al [11] and is popularly known as WBZ-α scheme Chung andHulbert [12] have combined Newmark, HHT-α, and WBZ-α schemes and developed a new schemepopularly known as Generalized-α scheme This scheme is second order accurate and unconditionallystable They also studied the stability and accuracy characteristics of the proposed scheme Zhou andZhou [36] proposed an implicit time integration scheme which has two control parameters to varythe accuracy To capture the high oscillatory modes accurately Liang [37] proposed a time integrationscheme 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

Dokain-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 ples In another time integration scheme by Gholampour and Ghassemieh [41], the approximation fordisplacement term is considered as fourth order polynomial with five coefficients They studied thecharacteristics of the proposed scheme for different damping and stiffness ratios Weighted residualintegration 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 previoustime steps respectively As several acceleration and velocity terms are included in the approximationfor displacement and velocity effects of local and residual errors are reduced Chang [44] discussed anew family of structure-dependant methods (SDM-2) and compared it with SDM-1 No overshoot indisplacement and velocity for SDM-2 is observed Also, SDM-2 was found to be computationally moreefficient than SDM-1

exam-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 Themethod combines the trapezoidal rule and the three-point backward Euler scheme to yield a compositescheme for numerical integration of nonlinear dynamical system of equations The method, unlike fore.g., Newmark scheme, has no parameter to choose or adjust The method is shown to be second orderaccurate 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 methoddoes not directly impose energy and momentum conservation Dong [47] has presented various timeintegration algorithms of second order accuracy based on a general four-step scheme that resemblesthe backward differentiation formulas An extension to the composite strategy of Bathe [45, 46] isproposed

Recently, Silva and Bezerra [3] have proposed a scheme which is based on the Bathe campsitescheme [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] havefurther analyzed the Bathe composite scheme for variable step sizes and have shown that at a particularvalue of the step size the period elongation is minimum and the numerical dissipation is maximum

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 proposedscheme is discussed in detail in Chapter 3

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

time t = t n + 1 in terms of the approximation to the solution at the current step time t = t n + 1[29].The first-order backward Euler scheme is given by

˙un + 1 = un + 1 − un

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 ˙unhave 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 t n+1(Eq 1.1), wehave :

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, ˙unand ¨un

2andβ > 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

4 andγ = 1

2, Newmark scheme givesaverage acceleration scheme, which is implicit, second order-accurate and unconditionallystable

Linear acceleration scheme: Forβ = 1

6 andγ = 1

2, Newmark scheme yields linear ation scheme, which is implicit and conditionally stable

scheme, which is implicit and conditionally stable In absence of viscous damping, thisscheme is fourth-order accurate

2, 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 integrationscheme and widely used when the time step restriction is not too severe, such as in elasticwave 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 t n to

t n + 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 (F n + 1 − F n) (2.19)Here,θ is a free parameter which controls the stability and accuracy of the algorithm [29] Theapproximations for displacement and velocity can be written as

the scheme is unconditionally stable For values θ > 2, numerical dissipation reduces andrelative period error increases Also, high overshooting behavior is a disadvantage in Wilson-θscheme [29]

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 t n+1 , Houbolt scheme uses equation of motion at t n+1 and two backwarddifference formulae, which is obtained from a cubic polynomial passing through four successive

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

M ¨un + 1 + C ˙un + 1 + K un + 1 = Fn + 1 (2.21)

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

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

h2(2 un+1 − 5 un + 4 un−1 − un−2)

schemes An important disadvantage of Houbolt scheme is, it has excessive algorithmic dampingand affect the low frequency modes too strongly Also, there is no parametric control overalgorithmic 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 freeparameter α rather than by the time step size so that it does not affect the lower modes toostrongly 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 equilibriumequation (Eq 1.1) is written in a modified form as

M ¨un + 1 + (1 + α) C ˙un + 1 − α C ˙un + (1 + α) K un + 1 − K un = Ft n+ α (2.23)where the parameterα is used to vary the numerical dissipation of the scheme, and t n + 1 +α =(1 + α) t n + 1 − α t n = t n + 1 + α h ,

2) 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 indisplacement and first-order overshoot in velocity [29]

dis-sipation parameter [11] This scheme retains the Newmark’s scheme approximation equationsfor displacement and velocity i.e., Eqs.( 2.16- 2.15) Using dynamic equilibrium equations at time

level t n + 1(Eq 1.1), we have

(1 − αB) M ¨un + 1 +αBM ¨un + C ˙un + 1 + K un + 1 = Fn + 1 (2.27)whereαBis the algorithmic parameter

Remarks:

• ForαB = 0, WBZ scheme reduces to Newmark average acceleration scheme

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

(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

M ¨un + 1 −αm + C ˙un + 1 −αf + K un + 1 −αf = Fn + 1 −αf, (2.31)where

un + 1 −αf = (1 − αf) un + 1 + αfun, (2.32)

˙un + 1 −αf = (1 − αf) ˙un + 1 + αf ˙un, (2.33)

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

tn + 1 −αf = (1 − αf) tn + 1 + αftn (2.35)

Remarks:

• Forαm = αf = 0, the scheme reduces to Newmark scheme [4]

• Forαm = 0 andαf = 0, the scheme reduces to HHT-α scheme and WBZ scheme respectively

• Conditions for unconditional stability are

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

t n +1 (Eq 1.1), we have :

M ¨un +1 + C ˙un +1 + K un +1 = Fn +1 (2.39)

Considering t n +1 =t n+ h2, Trapezoidal scheme is applied over the first substep The

approxima-tions for velocity and displacement at time t n +1 for Trapezoidal scheme are given by

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

The approximations for velocity and displacement at time t n + 1for three-point backward Eulerscheme are given by

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 t n +γt = t n +γt h

as an instance of time between t n and t n+1forγt ∈ (0, 1), trapezoidal scheme is applied over thefirst substep,γt h Using dynamic equilibrium equations at time level t n +γt(Eq 1.1), we have

M ¨un +γt + C ˙un +γt + N (u, t n +γt) = Fn +γt, (2.44)

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

which is, in general, a function of displacement vector u and time t, and F n +γt 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 n +γt) can be written as K u

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

scheme are given by

M ¨un + 1 + C ˙un +1 + N (u, t n + 1) = Fn + 1 (2.47)The approximations for velocity and acceleration for three-point backward Euler scheme aregiven by

˙un + 1 = c1un + c2un +γt + c3un + 1, (2.48)

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

where the constants c1, c2, and c3can be expressed as

Gohlam-pour et al [38–40] for direct time integration for non-linear dynamic problems In order to improvethe accuracy of the composite scheme, the order of acceleration was increased by including moreterms of the Taylor series Two parametersαgandδ 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 aregiven by

ut + h = ut + h ˙u t + h2

2 ¨ut +

h36

The value of the vectors utand utare computed using following approximation

• Relative period error in this scheme is similar compared to Newmark’s average accelerationand Generalised-α schemes but lesser than Wilson-θ scheme

and unconditionally stable scheme This scheme is similar to Houbolt scheme The dynamicequilibrium equation is same as Eq (2.21) Approximations for velocity and acceleration, however,

at time t n + 1are given as

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 stabilityand 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 h cr If large time steps (greater

than the critical time step h cr) are used, the numerical solution blows off and it diverges completelyfrom the actual solution Explicit schemes are widely used for fast transient analysis, for example, inthe analysis of crash problems

On the other hand in the implicit schemes, the displacement and the velocity at the current timestep are expressed not only interms of the values of the previous time step but also of the currenttime step Hence, the solution of system of resulting equations requires an iterative scheme, usuallyNewton-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.

Chapter 3

Proposed Time Integration Scheme

In the present chapter, an extension to the composite scheme proposed by Silva and Bezerra [3] ispreseneted In the proposed extension too the variable substep sizes are used However, the proposedimplicit composite scheme is a parameter based time integration scheme in which the Newmarkscheme [4] is applied in the first substep and three-point backward Euler scheme for the secondsubstep The composite scheme is shown schematically in Figure (3.1)

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

The governing equations of equilibrium for nonlinear transient structural dynamic problems is pressed as follows:

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

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

for displacement and velocity at time t n +γt for Newmark scheme are given by

(γt h)2

2

h(1 − 2β) ¨un + ( 2β ) ¨un +γt

i

whereβ, γ are Newmark scheme parameters After rearrangement, the acceleration and velocities at

time t n +γt can be written as

Euler scheme is given by

¨un + 1 = c1 ˙un + c2 ˙un +γt + c3c1un + c3c2un +γt +c23un + 1 (3.17)

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

M ¨un + 1 + C ˙un + 1 + N (u, t) n + 1 = Fn + 1 (3.18)Substituting for acceleration from Eq (3.17) we obtain

effective stiffness matrix K (ui

n + 1 ) at iteration i is obtained which is deformation-dependent The

expression for K (ui

n + 1) is given by

K (ui n + 1 ) = c23M + Kt(un + 1), (3.21)where

Kt(un +γt) = ∂ Nn + 1

The effective iterative equation is given by

K (ui n + 1)∆ ui = − Rn + 1 (3.23)

The displacements are updated as

ui + 1 n + 1 = ui n + 1 + ∆ ui (3.24)

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, fornonlinear dynamic problems, the trapezoidal scheme does not guarantee the conservation of energyand momentum as time progresses [18, 26, 45, 51] It fails to provide high frequency dissipation innonlinear analysis Even if smaller time step is considered convergence is not guaranteed as it maylead to excitation of even higher frequencies which lead to instability One of the earliest work on thespectral stability and accuracy analysis of direct time integration schemes has been done by Bathe andWilson [52] Also, Bathe [2] has discussed the stability and accuracy characteristics of several direct timeintegration 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 necessarycondition [26] Numerical dissipation is considered to be advantageous as it ensures better numericalstability 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 theintegration 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