Runge – Kuta (RK) method is reasonably simple and robust for numerical solution of differential equations but it requires an intelligent adaptive step-size routine; to achieve this, there is need to develop a good logical computer code. This study develops a finite element code in Java using Runge-Kuta method as a solution algorithm to predict dynamic time response of structural beam under impulse load. The solution obtained using direct integration and the present work is comparable.
IJERA (International journal of Engineering Research and Applications) is International online, ... peer reviewed journal. For more detail or submit your article, please visit www.ijera.com
International journal of engineering and mathematical modelling vol2 no1_2015_2IJEMM
This paper is devoted to the homogenization of the Maxwell equations with periodically oscillating coefficients in the bianisotropic media which represents the most general linear media. In the first time, the limiting homogeneous constitutive law is rigorously justified in the frequency domain and is found from the solution of a local problem on the unit cell. The homogenization process is based on the two-scale convergence conception. In the second time, the implementation of the homogeneous
constitutive law by using the finite element method and the introduction of the boundary conditions in the discrete problem are introduced. Finally, the numerical results associated of the perforated chiral media are presented.
The peer-reviewed International Journal of Engineering Inventions (IJEI) is started with a mission to encourage contribution to research in Science and Technology. Encourage and motivate researchers in challenging areas of Sciences and Technology.
IJERA (International journal of Engineering Research and Applications) is International online, ... peer reviewed journal. For more detail or submit your article, please visit www.ijera.com
International journal of engineering and mathematical modelling vol2 no1_2015_2IJEMM
This paper is devoted to the homogenization of the Maxwell equations with periodically oscillating coefficients in the bianisotropic media which represents the most general linear media. In the first time, the limiting homogeneous constitutive law is rigorously justified in the frequency domain and is found from the solution of a local problem on the unit cell. The homogenization process is based on the two-scale convergence conception. In the second time, the implementation of the homogeneous
constitutive law by using the finite element method and the introduction of the boundary conditions in the discrete problem are introduced. Finally, the numerical results associated of the perforated chiral media are presented.
The peer-reviewed International Journal of Engineering Inventions (IJEI) is started with a mission to encourage contribution to research in Science and Technology. Encourage and motivate researchers in challenging areas of Sciences and Technology.
System for Prediction of Non Stationary Time Series based on the Wavelet Radi...IJECEIAES
This paper proposes and examines the performance of a hybrid model called the wavelet radial bases function neural networks (WRBFNN). The model will be compared its performance with the wavelet feed forward neural networks (WFFN model by developing a prediction or forecasting system that considers two types of input formats: input9 and input17, and also considers 4 types of non-stationary time series data. The MODWT transform is used to generate wavelet and smooth coefficients, in which several elements of both coefficients are chosen in a particular way to serve as inputs to the NN model in both RBFNN and FFNN models. The performance of both WRBFNN and WFFNN models is evaluated by using MAPE and MSE value indicators, while the computation process of the two models is compared using two indicators, many epoch, and length of training. In stationary benchmark data, all models have a performance with very high accuracy. The WRBFNN9 model is the most superior model in nonstationary data containing linear trend elements, while the WFFNN17 model performs best on non-stationary data with the non-linear trend and seasonal elements. In terms of speed in computing, the WRBFNN model is superior with a much smaller number of epochs and much shorter training time.
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Hysteresis Loops for Magnetoelectric Multiferroics Using Landau-Khalatnikov T...IJECEIAES
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In this paper, we introduced the modified differential transform which is a modified version of a two-dimensional differential transform method. First, the properties of the modified differential transform method (MDTM) are presented. After this, by using the idea modified differential transform method we will find an analytical-numerical solution of linear partial integro-differential equations (PIDE) with convolution kernel which occur naturally in various fields of science and engineering. In some cases, the exact solution may be achieved. The efficiency and reliability of this method are illustrated by some examples.
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which is obtained by studying the possibility of building a hybrid solar receiver; It is necessary to mention that
the solution of the aforementioned system is relatively difficult to obtain through iterative methods since the
system is apparently unstable. To find this possible solution is used a novel numerical method valid for one and
several variables, which using the fractional derivative, allows us to find solutions for some nonlinear systems in
the complex space using real initial conditions, this method is also valid for linear systems. The method described
above has an order of convergence (at least) linear, but it is easy to implement and it is not necessary to invert
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We consider the generic stability of optimal control problems governed by nonlinear impulsive evolution equations. Under perturbations of the right-hand side functions of the controlled system, the results of stability for the impulsive optimal control problems are proved given set-valued theory.
International Journal of Engineering Research and Applications (IJERA) is an open access online peer reviewed international journal that publishes research and review articles in the fields of Computer Science, Neural Networks, Electrical Engineering, Software Engineering, Information Technology, Mechanical Engineering, Chemical Engineering, Plastic Engineering, Food Technology, Textile Engineering, Nano Technology & science, Power Electronics, Electronics & Communication Engineering, Computational mathematics, Image processing, Civil Engineering, Structural Engineering, Environmental Engineering, VLSI Testing & Low Power VLSI Design etc.
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In this paper, a novel method called Laplace-differential transform method (LDTM) is used to obtain an
approximate analytical solution for strong nonlinear initial and boundary value problems associated in
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an excellent agreement is demonstrated and discussed between the approximate solution and the exact one in
three examples. The most significant features of this method are its capability of handling non-linear boundary
value problems.
Finite Element Analysis Lecture Notes Anna University 2013 Regulation NAVEEN UTHANDI
One of the most Simple and Interesting topics in Engineering is FEA. My work will guide average students to score good marks. I have given you full package which includes 2 Marks and Question Banks of previous year. All the Best
For Guidance : Comment Below Happy to Teach and Learn along with you guys
International Journal of Engineering Research and Applications (IJERA) is an open access online peer reviewed international journal that publishes research and review articles in the fields of Computer Science, Neural Networks, Electrical Engineering, Software Engineering, Information Technology, Mechanical Engineering, Chemical Engineering, Plastic Engineering, Food Technology, Textile Engineering, Nano Technology & science, Power Electronics, Electronics & Communication Engineering, Computational mathematics, Image processing, Civil Engineering, Structural Engineering, Environmental Engineering, VLSI Testing & Low Power VLSI Design etc.
In this paper, we make use of the fractional differential operator method to find the modified Riemann-Liouville (R-L) fractional derivatives of some fractional functions include fractional polynomial function, fractional exponential function, fractional sine and cosine functions. The Mittag-Leffler function plays an important role in our article, and the fractional differential operator method can be applied to find the particular solutions of non-homogeneous linear fractional differential equations (FDE) with constant coefficients in a unified way and it is a generalization of the method of finding particular solutions of classical ordinary differential equations. On the other hand, several examples are illustrative for demonstrating the advantage of our approach and we compare our results with the traditional differential calculus cases.
System for Prediction of Non Stationary Time Series based on the Wavelet Radi...IJECEIAES
This paper proposes and examines the performance of a hybrid model called the wavelet radial bases function neural networks (WRBFNN). The model will be compared its performance with the wavelet feed forward neural networks (WFFN model by developing a prediction or forecasting system that considers two types of input formats: input9 and input17, and also considers 4 types of non-stationary time series data. The MODWT transform is used to generate wavelet and smooth coefficients, in which several elements of both coefficients are chosen in a particular way to serve as inputs to the NN model in both RBFNN and FFNN models. The performance of both WRBFNN and WFFNN models is evaluated by using MAPE and MSE value indicators, while the computation process of the two models is compared using two indicators, many epoch, and length of training. In stationary benchmark data, all models have a performance with very high accuracy. The WRBFNN9 model is the most superior model in nonstationary data containing linear trend elements, while the WFFNN17 model performs best on non-stationary data with the non-linear trend and seasonal elements. In terms of speed in computing, the WRBFNN model is superior with a much smaller number of epochs and much shorter training time.
Finite element modelling of nonlocal dynamic systems, Modal analysis of nonlocal dynamical systems, Dynamics of damped nonlocal systems, Numerical illustrations
Hysteresis Loops for Magnetoelectric Multiferroics Using Landau-Khalatnikov T...IJECEIAES
We present a theoretical discussion of the hysteresis in magnetoelectric multiferroics with bi-quadratic magnetoelectric coupling. The calculations were performed by employing Landau-Khalatnikov equation of motion for both the ferroelectric and ferromagnetic phase, then solve it simultaneously. In magnetoelectric, we obtain four types of hysteresis: ferroelectric hysteresis, ferromagnetic hysteresis and two types of cross hysteresis (electric field versus magnetization and magnetic field versus electric polarization). The cross hysteresis has butterfly shape which agree with the result from the previous research. It can also be seen from that hysteresis, that magnetization / electric polarization can not be flipped into the opposite direction using external electric / magnetic field when the magnetoelectric coupling is bi-quadratic type. Overall, the result shows that LandauKhalatnikov equation is able to approximate hysteresis loops in multiferroics system.
In this paper, we introduced the modified differential transform which is a modified version of a two-dimensional differential transform method. First, the properties of the modified differential transform method (MDTM) are presented. After this, by using the idea modified differential transform method we will find an analytical-numerical solution of linear partial integro-differential equations (PIDE) with convolution kernel which occur naturally in various fields of science and engineering. In some cases, the exact solution may be achieved. The efficiency and reliability of this method are illustrated by some examples.
CHN and Swap Heuristic to Solve the Maximum Independent Set ProblemIJECEIAES
We describe a new approach to solve the problem to find the maximum independent set in a given Graph, known also as Max-Stable set problem (MSSP). In this paper, we show how Max-Stable problem can be reformulated into a linear problem under quadratic constraints, and then we resolve the QP result by a hybrid approach based Continuous Hopfeild Neural Network (CHN) and Local Search. In a manner that the solution given by the CHN will be the starting point of the local search. The new approach showed a good performance than the original one which executes a suite of CHN runs, at each execution a new leaner constraint is added into the resolved model. To prove the efficiency of our approach, we present some computational experiments of solving random generated problem and typical MSSP instances of real life problem.
Fractional pseudo-Newton method and its use in the solution of a nonlinear sy...mathsjournal
The following document presents a possible solution and a brief stability analysis for a nonlinear system,
which is obtained by studying the possibility of building a hybrid solar receiver; It is necessary to mention that
the solution of the aforementioned system is relatively difficult to obtain through iterative methods since the
system is apparently unstable. To find this possible solution is used a novel numerical method valid for one and
several variables, which using the fractional derivative, allows us to find solutions for some nonlinear systems in
the complex space using real initial conditions, this method is also valid for linear systems. The method described
above has an order of convergence (at least) linear, but it is easy to implement and it is not necessary to invert
some matrix for solving nonlinear systems and linear systems.
Stability analysis for nonlinear impulsive optimal control problemsAI Publications
We consider the generic stability of optimal control problems governed by nonlinear impulsive evolution equations. Under perturbations of the right-hand side functions of the controlled system, the results of stability for the impulsive optimal control problems are proved given set-valued theory.
International Journal of Engineering Research and Applications (IJERA) is an open access online peer reviewed international journal that publishes research and review articles in the fields of Computer Science, Neural Networks, Electrical Engineering, Software Engineering, Information Technology, Mechanical Engineering, Chemical Engineering, Plastic Engineering, Food Technology, Textile Engineering, Nano Technology & science, Power Electronics, Electronics & Communication Engineering, Computational mathematics, Image processing, Civil Engineering, Structural Engineering, Environmental Engineering, VLSI Testing & Low Power VLSI Design etc.
A Mathematical Model to Solve Nonlinear Initial and Boundary Value Problems b...IJERA Editor
In this paper, a novel method called Laplace-differential transform method (LDTM) is used to obtain an
approximate analytical solution for strong nonlinear initial and boundary value problems associated in
engineering phenomena. It is determined that the method works very well for the wide range of parameters and
an excellent agreement is demonstrated and discussed between the approximate solution and the exact one in
three examples. The most significant features of this method are its capability of handling non-linear boundary
value problems.
Finite Element Analysis Lecture Notes Anna University 2013 Regulation NAVEEN UTHANDI
One of the most Simple and Interesting topics in Engineering is FEA. My work will guide average students to score good marks. I have given you full package which includes 2 Marks and Question Banks of previous year. All the Best
For Guidance : Comment Below Happy to Teach and Learn along with you guys
International Journal of Engineering Research and Applications (IJERA) is an open access online peer reviewed international journal that publishes research and review articles in the fields of Computer Science, Neural Networks, Electrical Engineering, Software Engineering, Information Technology, Mechanical Engineering, Chemical Engineering, Plastic Engineering, Food Technology, Textile Engineering, Nano Technology & science, Power Electronics, Electronics & Communication Engineering, Computational mathematics, Image processing, Civil Engineering, Structural Engineering, Environmental Engineering, VLSI Testing & Low Power VLSI Design etc.
In this paper, we make use of the fractional differential operator method to find the modified Riemann-Liouville (R-L) fractional derivatives of some fractional functions include fractional polynomial function, fractional exponential function, fractional sine and cosine functions. The Mittag-Leffler function plays an important role in our article, and the fractional differential operator method can be applied to find the particular solutions of non-homogeneous linear fractional differential equations (FDE) with constant coefficients in a unified way and it is a generalization of the method of finding particular solutions of classical ordinary differential equations. On the other hand, several examples are illustrative for demonstrating the advantage of our approach and we compare our results with the traditional differential calculus cases.
International Journal of Engineering Research and Applications (IJERA) is an open access online peer reviewed international journal that publishes research and review articles in the fields of Computer Science, Neural Networks, Electrical Engineering, Software Engineering, Information Technology, Mechanical Engineering, Chemical Engineering, Plastic Engineering, Food Technology, Textile Engineering, Nano Technology & science, Power Electronics, Electronics & Communication Engineering, Computational mathematics, Image processing, Civil Engineering, Structural Engineering, Environmental Engineering, VLSI Testing & Low Power VLSI Design etc.
A finite element modelling of composite plate with
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active vibration control is presented in this paper. The
displacement feedback (DF) and direct velocity feedback (DVF)
controls are integrated into the FE software ANSYS to perform
closed loop analysis for vibration control. A smart laminated
composite beam with different layup configurations under free
and forced vibration condition is studied and the results shows
suppression of vibration achieved successfully in both DF and
DVF controls.
IJERA (International journal of Engineering Research and Applications) is International online, ... peer reviewed journal. For more detail or submit your article, please visit www.ijera.com
DETAILED STUDIES ON STRESS CONCENTRATION BY CLASSICAL AND FINITE ELEMENT ANAL...IAEME Publication
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Integration of Finite Element Method with Runge – Kuta Solution Algorithm
1. Olawale Simon .et.al. Int. Journal of Engineering Research and Application www.ijera.com
ISSN : 2248-9622, Vol. 7, Issue 4, ( Part -6) April 2017, pp.45-50
www.ijera.com DOI: 10.9790/9622-0704064650 45 | P a g e
Integration of Finite Element Method with Runge – Kuta Solution
Algorithm
Olawale Simon, Ogunbiyi Moses A. Alabi Olusegun and Ofuyatan Olatokunbo
1
Department of Civil Engineering Faculty of Engineering and Environmental Sciences Osun State University,
Osogbo, Nigeria
2
Department of Civil Engineering Covenant University, Ota Ogun State
ABSTRACT
Runge – Kuta (RK) method is reasonably simple and robust for numerical solution of differential equations but
it requires an intelligent adaptive step-size routine; to achieve this, there is need to develop a good logical
computer code. This study develops a finite element code in Java using Runge-Kuta method as a solution
algorithm to predict dynamic time response of structural beam under impulse load. The solution obtained using
direct integration and the present work is comparable.
I. INTRODUCTION
In numerical analysis, the Runge-Kuta
method is a family of implicit and explicit iterative
methods, which includes the well – known routine
called Euler methods, used in temporal
discretization for the approximate solution of
Ordinary Differential Equation (ODE) (Devries and
Hasbun, 2011). Runge-Kuta method is reasonably
simple and robust and is a good candidate for
numerical solution of differential equations when
combined with an intelligent adaptive step-size
routine (Abramowitz and Stegun, 1972).The
Runge-Kuta Algorithm is known to be very
accurate and well – behaved for a wide range of
problems but to describe it precisely we need to
develop some notation and a good logical computer
code; which this study endeavored to achieve.
II. THEORETICAL BACKGROUND
Finite Element Analysis (FEA) is a branch
of solid mechanics which can be applied to solve
multi-physics problems. Its applications include
structural analyses, solid mechanics, dynamics,
thermal analysis, electrical analysis and
biomaterials (Hughes, 1987 and Logan, 2002). The
major purpose of FEA is to determine the values of
the displacements, stresses and strains at each
material point if a force is applied on a solid (Jerry,
2006).
The Runge-Kuta algorithm works over time step
increment to implicitly calculate the responses over
time domain, starting from the initial time t0 to the
time limit tmax.
Methodology: Study Solution Development
The equation of motion in single degree of freedom (SDF) is given by
1
and the displacement equation in terms of shape functions and time is given by
u(x,t) = [ ] u(t) 2
or u(x,t) = [A] u(t) and the shape functions are defined as follows :
=
= 3
=
=
And u(t) is the nodal displacement at time t
External forces: g(x,t) = 4
and
f (x,t) is the applied force
By the principle of virtual works:
5
RESEARCH ARTICLE OPEN ACCESS
2. S. Dewangan.et.al. Int. Journal of Engineering Research and Application www.ijera.com
ISSN : 2248-9622, Vol. 7, Issue 4, ( Part -4) April 2017, pp.45-
www.ijera.com DOI: 10.9790/9622-0704064650 46 | P a g e
6
7
where
8
9
10
Equating 9 and 10
11
Where the consistent matrices of mass, stiffness, damping and force are given below
Runge-Kuta Method of Solution
The solution to the equation of motion can
be obtained using Runge-Kuta (RK) method which
very suited to initial condition system. However,
the integration of Finite Element Method with RK
method requires some careful of considerations
because the overall global U vector is a
combination of displacement and velocity vectors.
The RK solution decomposes the equation of
motion into two equations U1 = U and U2 = dU1/dt.
Thus the initial conditions to start the solution
procedure are given below. Please note that U is
the combination of global displacement and
velocity and is different from u.
U1 = 0 (U1pre ), U2 = 0 (U2pre ) at t = 0
=
(2,8) matrix 12
3. S. Dewangan.et.al. Int. Journal of Engineering Research and Application www.ijera.com
ISSN : 2248-9622, Vol. 7, Issue 4, ( Part -4) April 2017, pp.45-
www.ijera.com DOI: 10.9790/9622-0704064650 47 | P a g e
)
)
.
Infact, where N x N is the size of global consistent stiffness, damp and mass matrices
Pseudo Code
Step 1: Calculate the member stiffness matrix [K]4x4 , mass matrix [M]4x4and damping matrix
[C]4x4= β [M]4x4
Step 2: Set start time t[0] = tini
Calculate the time step dt = , n being the total steps
Step 2: Set up [U]initial and set [U]i-1 = [U]initial
Step 3: Set time t[i] = t [i-1] + dt
Step 4: Assemble the global stiffness matrix NxNK , mass matrix NxNM and damping
matrix NxNC = β NxN
M
Step 5: Compute x
Step 6: Compute
( )
([𝑖−1]+ , [ 𝑈 ]𝑖+ )
)
( ) / 6.0
Step 7: Extract global displacement, velocities and
compute acceleration which are N x 1 size.
Step 8: Increase time to t [i] = t [i -1] + dt and
repeat Step 5 to Step 7.
III. RESULTS AND DISCUSSION
This study tests the present solution of the
equation of motion by analyzing a prismatic
concrete beam of 200 x 200 mm cross section by
3.0m length. The study used material characteristic
of Young’s modulus of 48.39 MPa and yield stress
of 65.00 MPa. A triangular force excitation of
maximum value of 500KN, decaying to zero on the
positive phase of 0.015 ms was applied over a time
domain. The same problem was analyzed using
Direct Integration and Runge-Kuta methods for
both damped and un-damped situations. The results
of the comparison of the two methods are shown
below in figures 1 and 2 respectively.
The agreement between the two methods
is reasonable and indicates that Runge-Kuta
method integrated with Finite Element Method can
result in accurate prediction of the time response of
structural elements over the period of excitation.
With more attention paid to details, the two
methods can seamlessly converge to the same
solution with practically no difference.
4. S. Dewangan.et.al. Int. Journal of Engineering Research and Application www.ijera.com
ISSN : 2248-9622, Vol. 7, Issue 4, ( Part -4) April 2017, pp.45-
www.ijera.com DOI: 10.9790/9622-0704064650 48 | P a g e
Figure 1: Comparison of Runge-Kuta Method with Direct Integration Method for damped motion.
Figure 2: Comparison of Runge-Kuta Method with Direct Integration Method for un-damped motion.
IV. CONCLUSION
The agreement between the two methods
is reasonable and indicates that Runge-Kuta
method integrated with finite element method can
result in accurate prediction of the time response of
structural elements over the period of excitation.
REFERENCES
[1]. Abramowtiz, M. and Stegun, I.A. (1972)
Handbook of Mathematical foundation with
formulas, Graphs and Mathematical tables
9th
Edition, New York: Dover pp. 896-897.
[2]. Barthe K.J. (1996) The finite element
procedures. Prentice Hall.
[3]. Devries, P.L. and Hasbun, J.E. (2011) A first
course in Computational Physics (2nd
Edition) Jones and Bartlett Publishers, pg.
215.
[4]. Huges, T.J.R. (1987) The finite element
methods: Linear static and dynamic finite
element analysis. Dove publication.
[5]. Jerry, H.Q. (2006) Finite element analysis
note book.
[6]. Logen, D.L.(2002) A first course in finite
element (3rd
Edition)
Appendix
Although the detailed listing of the Java
code may be required by some inquisitive readers,
effort is made to provide the Javadoc listings below
to assist in recreating the code quickly.
Java Code Definitions:
Class DynaBeamRK
java.lang.Object
o DynaBeamRK
public class DynaBeamRK
extends java.lang.Object
o Constructor Summary
Constructors
Constructor and Description
DynaBeamRK(int numberEleme
n, float timeLimit,
int numberOfTimeStep)
o Method Summary
Methods
Modifi
er and
Type
Method and
Description
static
void
calcK1(int step,
float deltaTime)
5. S. Dewangan.et.al. Int. Journal of Engineering Research and Application www.ijera.com
ISSN : 2248-9622, Vol. 7, Issue 4, ( Part -4) April 2017, pp.45-
www.ijera.com DOI: 10.9790/9622-0704064650 49 | P a g e
static
void
calcK2(int step,
float deltaTime)
static
void
calcK3(int step,
float deltaTime)
static
void
calcK4(int step,
float deltaTime)
static
void
calcU(int step)
static
void
calcU0()
static
void
calcU01()
static
void
calcU02()
static
void
cofactor(float[][] num)
static
void
computeAcceleration(i
nt step)
static
void
computeElementMatri
x()
static
void
computeElemForces(int
t)
static
void
computeForce(int step,
float addedT)
static
void
computeNodalAccel(int
t)
static
void
computeNodalDisp(int t
)
static
void
computeNodalVel(int t)
static
void
computeTimeDispHisto
ry()
static
void
computeTimeRespHist
RK()
static
float
determinant(float[][] nu
m, int s)
static
void
initialise()
static
void
initialiseIntermediate()
static
void
main(java.lang.String[]
args)
static
void
readBasicInput()
static
void
readInputData()
static
void
transpose(float[][] num)
Methods inherited
from
class java.lang.Object
clone, equals, finalize,
getClass, hashCode,
notify, notifyAll,
toString, wait, wait, wait
o Constructor Detail
DynaBeamRK
public DynaBea
mRK(int numberElemen
,
float timeLimit,
int numberOfTimeStep)
o Method Detail
calcU01
public
static void calcU01()
calcU02
public
static void calcU02()
calcU0
public
static void calcU0()
initialiseIntermediate
public
static void initialiseInter
mediate()
calcK1
public
static void calcK1(int ste
p,
float deltaTime)
calcK2
public
static void calcK2(int ste
p,
float deltaTime)
calcK3
public
static void calcK3(int ste
p,
float deltaTime)
calcK4
public
static void calcK4(int ste
p,
float deltaTime)
6. S. Dewangan.et.al. Int. Journal of Engineering Research and Application www.ijera.com
ISSN : 2248-9622, Vol. 7, Issue 4, ( Part -4) April 2017, pp.45-
www.ijera.com DOI: 10.9790/9622-0704064650 50 | P a g e
calcU
public
static void calcU(int step
)
readInputData
public
static void readInputDat
a()
readBasicInput
public
static void readBasicInp
ut()
initialise
public
static void initialise()
computeElementMatri
x
public
static void computeElem
entMatrix()
computeTimeDispHist
ory
public
static void computeTime
DispHistory()
computeTimeRespHist
RK
public
static void computeTime
RespHistRK()
computeNodalDisp
public
static void computeNod
alDisp(int t)
computeNodalVel
public
static void computeNod
alVel(int t)
computeNodalAccel
public
static void computeNod
alAccel(int t)
computeElemForces
public
static void computeElem
Forces(int t)
computeForce
public
static void computeForc
e(int step,
float addedT)
computeAcceleration
public
static void computeAcce
leration(int step)
determinant
public
static float determinant(f
loat[][] num,
int s)
cofactor
public
static void cofactor(float
[][] num)
transpose
public
static void transpose(flo
at[][] num)
main
public
static void main(java.lan
g.String[] args)