Dynamic Relaxation Method. Theoretical Analysis, Solved Examples and Computer Programming
©2016
Textbook
43 Pages
Summary
This book is suitable as a textbook for a first course on Dynamic Relaxation technique in civil and mechanical engineering curricula. It can be used as a reference by engineers and scientists working in the industrial sector and in academic institutions.
The first chapter includes an introduction to the Dynamic Relaxation method (DR) which is combined with the Finite Differences method (FD) for the sake of solving ordinary and partial differential equations, as a single equation or as a group of differential equations. In this chapter the dynamic relaxation equations are transformed to artificial dynamic space by adding damping and inertia effects. These are then expressed in finite difference form and the solution is obtained through iterations.
In the second chapter the procedural steps in solving differential equations using the DR method were applied to the system of differential equations (i.e. ordinary and/or partial differential equations). The DR program performs the following operations: Reads data file; computes fictitious densities; computes velocities and displacements; checks stability of numerical computations; checks convergence of solution; and checks wrong convergence. At the end of this chapter the Dynamic Relaxation numerical method coupled with the Finite Differences discretization technique is used to solve nonlinear ordinary and partial differential equations. Subsequently, a FORTRAN program is developed to generate the numerical results as analytical and/or exact solutions.
The first chapter includes an introduction to the Dynamic Relaxation method (DR) which is combined with the Finite Differences method (FD) for the sake of solving ordinary and partial differential equations, as a single equation or as a group of differential equations. In this chapter the dynamic relaxation equations are transformed to artificial dynamic space by adding damping and inertia effects. These are then expressed in finite difference form and the solution is obtained through iterations.
In the second chapter the procedural steps in solving differential equations using the DR method were applied to the system of differential equations (i.e. ordinary and/or partial differential equations). The DR program performs the following operations: Reads data file; computes fictitious densities; computes velocities and displacements; checks stability of numerical computations; checks convergence of solution; and checks wrong convergence. At the end of this chapter the Dynamic Relaxation numerical method coupled with the Finite Differences discretization technique is used to solve nonlinear ordinary and partial differential equations. Subsequently, a FORTRAN program is developed to generate the numerical results as analytical and/or exact solutions.
Excerpt
Table Of Contents
Contents
Dedication
i
Acknowledgement
ii
Preface iii
Contents
iv
Symbols and Abbreviations
v
Chapter One: Introduction
1.1 General Introduction
1
1.2 Formulation of Dynamic Relaxation Equations
1
1.3 Finite Difference Approximation
4
1.3.1 Ordinary Differential Equations
4
1.3.2 Partial Differential Equations
7
Chapter Two: Procedural Steps in Solving Differential Equations
Using Dynamic Relaxation (DR) Method
2.1 The Dynamic Relaxation (DR) Program
11
2.1.1 Numerical Instability
11
2.1.2 Convergence of DR Solution
11
2.1.3 Convergence to an Invalid Solution
12
2.1.4 Time Increment
12
2.1.5 Damping Coefficient
12
2.2 Solved Examples
12
2.2.1 Solution of Ordinary Differential Equations
13
2.2.2 Solution of Partial Differential Equations
17
Bibliography
31
iv
Symbols and Abbreviations
= K
*
= maximum number of iterations
= equal to
= discretization of solution in the x
direction
= greater than
= less than
= velocity
= acceleration
= time increment
= inertia effect
= damping effect
v
Chapter One
Introduction
1.1 General Introduction:
Dynamic Relaxation method (DR) Coupled with Finite Differences method
(FD) is used for solving ordinary and partial differential equations as a single
equation or as a group of differential equations. To apply dynamic relaxation
software technique, the differential equations are transformed into dynamic equations
by adding damping and inertia elements. These in turn are expressed in finite
differences form, and the solution is obtained by an iterative procedure as is
explained in the following paragraphs:
The differential equation is referred to in the following as:
= 0 (1.1)
Where, f = 0, may be an ordinary differential equation as follows:
( )
+ ( )
+ ( ) = 0
Or a partial differential equation as stated below:
( )
+ ( )
+ ( , ) = 0
1.2 Formulation of Dynamic Relaxation Equations:
The dynamic relaxation method (DR) formula begins with the dynamic equation
which may be written as:
=
+
(1.2)
In this procedure the statical system i.e. equation (
1.1) is transferred to an artificial
dynamic space by adding fictitious inertia and damping forces as in equation (1.2).
The DR method was first proposed in 1960s; refer to Rushton [1], Cassel and
Hobbs [2], and Day [3]. In this method, the equations of equilibrium are converted to
dynamic equations by adding damping and inertia terms, these are then expressed in
finite difference form and solution is obtained through iterations. The optimum
1
damping coefficient and time increment used to stabilize the solution depend on a
number of factors including the stiffness matrix of the structure, the applied load, the
boundary conditions and the size of the mesh used, etc.
In order to analyze various complicated problems in engineering, many kinds of
efficient numerical methods such as finite difference method, finite element method
and the weighted residual method have been developed. However, the accompanying
problem is that large computers are needed to solve the related large scale equations.
Sometimes, the equations are so large that one can only obtain rough results. This is
especially conspicuous in solving non
linear problems. In addition, numerical
instability during iteration is often involved.
In the traditional methods of solving equations from static equilibrium problems,
it is considered that internal forces exist initially in the structures. In so doing, one
assumes that the external forces were exerted very slowly so that the dynamic process
of the structures could be neglected. In fact, as has been pointed out by Rayleigh [4],
static solution of a mechanics system can be referred to as the steady state part of the
transient response of the system to step loading. This approach was successfully
applied to solving linear problems by Otter [5] and Day [3] in dependently in 1965,
and was named the dynamic relaxation (DR) method.
Nowadays, researchers are attracted by the efficiency of solving non
linear
problems with DR. The applications of DR to various problems indicate that the
method has the following distinctive features
^ see, for example, [6] [9] `.
Numerical techniques other than the dynamic relaxation (DR) method include
finite element method (FEM), which is widely used in most of the theoretical
analyses of today's research. In a comparison between the dynamic relaxation method
and the finite element method, Aalami [10] found that the computer time required for
finite element method is eight times greater than that for the dynamic relaxation
analysis, whereas storage capacity for finite element analysis is ten times or more
than that for DR analysis. This fact is supported by Putcha and Reddy [11], and
Turvey and Osman
^ [12] [14] `, who they noted that some of the finite element
2
formulations require large storage capacity and computer time. However, if the
analysis requires less computations and computer time, then, the dynamic relaxation
is considered more efficient than the finite element method. In another comparison
Aalami [10] found that the difference in accuracy between one version of finite
element and another may reach a value of 10% or more, whereas a comparison
between one version of finite element method and DR showed a difference of more
than 15%. Therefore, the dynamic relaxation method (DR) can be considered of
acceptable accuracy.
The only apparent limitation of dynamic relaxation (DR) method is that it can
only be applied to limited geometries. However, this limitation is irrelevant to square
and rectangular plates and beams which are widely used in engineering applications.
The errors inherent in the dynamic relaxation (DR) technique
^[15] [23]`
include discretization error which is due to the replacement of a continuous function
with a discrete function. Also, there is an additional error resulting from the non
exact solution of the discrete equations due to the variations of the velocities from the
edges of the plate to the center. The usage of finer meshes reduces the discretization
error, but increases the round
off error due to the large amount of computations
involved.
For the sake of simplifying and explanation of the DR method, in equation
(1.2) is referred to as displacement, and hence the terms
/
and
/
are the
velocity and acceleration respectively. Accordingly the first and second terms on the
right
hand side are the inertia and damping terms respectively. and are the
inertia and damping coefficients respectively, and is time.
If the velocities before and after the period
at an arbitrary node in the finite
difference mesh are denoted by
{ / }
and
{ / } respectively, then using
finite differences in time, and specifying the value of the function at
( - ), it is
possible to write equation (1.2) in the following form:
=
-
+
(1.3)
3
Now
, which is the velocity at the middle of the time increment, can be
approximated by the mean velocities before and after the time increment,
, which is
expressed as follows:
=
1
2
-
Hence, equation (1.3) can be expressed in the following form as:
=
-
+
2
-
(1.4)
Equation (1.4) can then be arranged to give the velocity after the time interval,
:
= (1 +
)
+ (1 -
)
(1.5)
Where:
=
2
The displacements at the middle of the next time increment can be determined by
integrating the velocity, so that:
=
+
(1.6)
The iterative procedure begins at time
= 0 with all initial values of the velocities
and displacements equal to zero or any other suitable values. In the first iteration, the
velocities are obtained from equation (1.5) and the displacements from equation
(1.6). The boundary conditions are then applied. Subsequent iterations follow the
same steps until the desired accuracy is achieved.
1.3 Finite Difference Approximation:
1.3.1 Ordinary Differential Equations:
The values of the interpolating function
( ) in the vicinity of the node in a
non
uniform or graded mesh shown in figure (1.1) below can be expressed as
follows using Taylor's series:
( + 1) = ( ) +
( ) +
2!
( )
4
+
3!
( ) +
4!
( ) + (1.7)
( - 1) = ( ) - ( ) +
2!
( )
-
3!
( ) +
4!
( ) + (1.8)
Figure (1.1) Non
uniform or graded mesh
Where
( ),
( ),
( ), and
( ) are the first, second, third, and fourth
derivatives of the function
( ) at node .
When multiplying equation (1.7) by
and equation (1.8) by
, then subtract the
latter from the former and rearrange the resulting expression to obtain the function at
node as follows:
( ) =
1
( )
( + 1) +
( )
( ) +
( )
( - 1) + (1.9)
Where:
( )
=
( +
)
5
( )
= -
-
( )
= -
( +
)
= -
6
+ (1.10)
Multiply equation (1.7) by and equation (1.8) by
and add them together to
obtain the second derivative of the function
( ) at node as follows:
( ) =
2
( )
( + 1) +
( )
( ) +
( )
( - 2) + (1.11)
Where:
( )
=
1
( +
)
( )
= -
1
( )
=
1
( +
)
=
-
3
-
-
+
12
+ (1.12)
If the derivatives of
( ) which are greater than the third are assumed negligible i.e.
the actual function approximates a quadratic function then
and represent the
error in the approximation resulting from replacing the actual function by a quadratic
function. The error in the first derivative of the function of equation (1.10) depends
on the graded mesh and it is proportional to
. The error in the second derivative
of the function, equation (1.12), is proportional to
for a uniform mesh (i.e.
=
), and proportional to
for a graded mesh (i.e.
). That is to say
the error associated with a graded mesh is greater than that of a uniform mesh with
the same number of elements. However, a graded mesh is more flexible than a
uniform mesh and it allows closer nodes to be employed in those regions where a
higher degree of accuracy is required.
When the mesh is uniform
=
, and hence:
6
Details
- Pages
- Type of Edition
- Erstausgabe
- Year
- 2016
- ISBN (PDF)
- 9783960675846
- ISBN (Softcover)
- 9783960670841
- File size
- 635 KB
- Language
- English
- Institution / College
- Nile Valley University – Faculty of Engineering & Technology
- Publication date
- 2016 (September)
- Keywords
- FORTRAN Dynamic Relaxation Differential Equation Finite Difference Approximation Civil engineering Mechanical engineering Guidebook
