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Static and dynamic analysis of two-dimensional elastic continua by combined finite difference with distributed transfer function method

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Static and dynamic analysis of two-dimensional elastic continua by combined finite difference with distributed transfer function method

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STATIC AND DYNAMIC ANALYSIS OF TWO-DIMENSIONAL ELASTIC
CONTINUA BY COMBINED FINITE DIFFERENCE WITH DISTRIBUTED
TRANSFER FUNCTION METHOD

by

Yau-Bin Yang

________________________________________________________________________

A Dissertation Presented to the
FACULTY OF THE GRADUATE SCHOOL
UNIVERSITY OF SOUTHERN CALIFORNIA
In Partial Fulfillment of the
Requirements for the Degree
DOCTOR OF PHILOSOPHY
(MECHANICAL ENGINEERING)

December 2007

Copyright 2007 Yau-Bin Yang
ii
ACKNOWLEDGMENTS

Many people have inspired, guided, and helped me during the long time I spent at
University of Southern California, and I would like to thank them all for a great graduate
school experience.
I present my immense gratitude to my dissertation advisor, Professor Bingen Yang for his
professional guidance, high standard, and support throughout my Ph. D. study.
I also acknowledge the members of my committee, Professor Flashner, Professor Sadhal,
Professor Shiflett, and Professor Lee. I got valuable advice for my dissertation from them.
Without their kind help, I can’t get involved in many fields in short time.
Many thanks for my friends. Some of them provide me the good resource on my research.
Some of them give me encouragements when I was in low depressed mood. Some of
them give me the new vision in my life.
I should thank my understanding wife, Chun-Chen and my lovely daughter Atina Yang.
With their spiritual support and encouragements, I finally can complete my whole
research. I also specially thank my parents, Hsin-Hsiung Yang and Yung-Ying Yang-
Huang who have supported me throughout the very long time study in financial and
spiritual support. And my two sisters, Huey-Tyng Yang and Yuan-Chuen Yang, they all
give me the support during my longest study.

iii
TABLE OF CONTENTS
Acknowledgements ii
List of Tables viii
List of Figures ix
Abstract xiii

Chapter 1 Introduction..................................................................................................... 1
1.1 Background and Motivation ..................................................................................... 1
1.2 Previous Research..................................................................................................... 2
1.3 Objectives and Scope of the Study ........................................................................... 7

Chapter 2 Basic Ideas ..................................................................................................... 10
2.1 Problem Statements ................................................................................................ 10
2.2 Background Theories.............................................................................................. 13
2.2.1 Elasticity .......................................................................................................... 13
2.2.2 Heat conduction in two-dimensional region.................................................... 15
2.2.3 Plate.................................................................................................................. 16
2.3 Boundary Conditions .............................................................................................. 18
2.3.1 Two-dimensional elasticity problems.............................................................. 18
2.3.1.1 Displacement boundary conditions........................................................... 18
2.3.1.2 Traction boundary conditions ................................................................... 18
2.3.1.3 Mixed boundary conditions ...................................................................... 19
2.3.2 Heat conduction in two-dimensional region.................................................... 19
2.3.2.1 Prescribed surface temperature................................................................. 19
2.3.2.2 Prescribed heat flux through the boundary surface................................... 19
2.3.2.3 Heat transfer by convection and radiation ................................................ 20
2.3.2.4 Contact between two solids ...................................................................... 20
2.3.3 Plate problems.................................................................................................. 20
2.3.3.1 Clamped side............................................................................................. 20
2.3.3.2 Simple support side................................................................................... 21
2.3.3.3 Free side.................................................................................................... 21
2.4 Several Important Equations in the polar coordinate Systems ............................... 21
2.5 Finite Difference Schemes...................................................................................... 23
2.6 State Space Formulation ......................................................................................... 24
2.7 Solution in Frequency Domain by Distributed Transfer Function Method............ 26
2.8 Solution in Time Domain by Distributed Transfer Function Method .................... 27
2.9 Flow Chart .............................................................................................................. 28

Chapter 3 Static and Dynamic Analysis of Two-dimensional Elasticity Problems .. 30
3.1 Problem Statement.................................................................................................. 30
3.2 Proposed Method in the Cartesian coordinate Systems.......................................... 30
3.2.1 Finite difference discretization in the Cartesian coordinate systems............... 30
3.2.2 Boundary conditions in the Cartesian coordinate systems .............................. 31
3.2.2.1 Left nodes.................................................................................................. 32
3.2.2.1.1 The line 0 ( ) x i = with traction boundary condition ........................... 32
3.2.2.1.2 The line 0 ( ) x i = with clamped boundary conditions......................... 33
3.2.2.1.3 The line 0 ( ) x i = with mixed boundary conditions ............................ 33
3.2.2.2 Right nodes ............................................................................................... 33
3.2.2.2.1 The procedure on () x L iii = is similar with the left nodes............... 33
3.2.2.3 Bottom mesh lines..................................................................................... 34
3.2.2.3.1 The line 0 ( ) y ii = with traction boundary condition.......................... 34
3.2.2.3.2 The line 0 ( ) y ii = with clamped boundary conditions....................... 35
3.2.2.3.3 The line 0 ( ) y ii = with mixed boundary condition............................ 36
3.2.2.4 Upper mesh lines....................................................................................... 36
3.2.2.4.1 The line () y Hiv = with traction boundary condition........................ 36
3.2.2.4.2 The line () y Hiv = with clamped boundary condition ...................... 37
3.2.2.4.3 The line () y Hiv = with mixed boundary condition......................... 37
3.3 Proposed Method in the polar coordinate systems ................................................. 38
3.3.1 Finite difference discretization in the polar coordinate systems...................... 38
3.3.2 Boundary conditions in the polar coordinate systems ..................................... 39
3.3.2.1 Right nodes ............................................................................................... 40
3.3.2.1.1 The line 0 ( ) iii θ = with traction boundary condition......................... 40
3.3.2.1.2 The line 0 ( ) iii θ = with clamped boundary condition ....................... 40
3.3.2.1.3 The line 0 ( ) iii θ = with mixed boundary conditions ......................... 40
3.3.2.2 Left nodes.................................................................................................. 41
3.3.2.2.1 The procedure on line /2 ( ) i θ π = is similar with the right nodes.... 41
3.3.2.3 Bottom mesh lines..................................................................................... 41
3.3.2.3.1 The arc
0
() rr ii = with Traction boundary condition ......................... 41
3.3.2.3.2 The arc
0
() rr ii = with clamped boundary condition ......................... 42
3.3.2.3.3 The arc
0
() rr ii = with mixed boundary condition............................. 43
3.3.2.4 Upper mesh lines....................................................................................... 43
3.3.2.4.1 The arc
0
() rr H iv = + with traction boundary condition................... 43
3.3.2.4.2 The arc
0
() rr H iv = + with clamped boundary condition ................. 44
3.3.2.4.3 The arc
0
() rr H iv = + with mixed boundary condition..................... 44
3.4 Assembly of Governing Equations and Boundary Conditions............................... 45
3.5 State Space Formulations........................................................................................ 46
3.6 Solution by DTFM.................................................................................................. 48
3.7 Assembly of Multiple Rectangular Subsections..................................................... 48
3.8 Static Numerical Examples..................................................................................... 50
3.9 Dynamic Numerical examples................................................................................ 74
3.10 Discussion............................................................................................................. 76

iv
Chapter 4 Steady and Unsteady State Heat Conduction Problems in Two-
dimensional Region......................................................................................................... 78
4.1 Problem Statement.................................................................................................. 78
4.2 Proposed Method in the Cartesian coordinate Systems.......................................... 78
4.2.1 Finite difference discretization in the Cartesian coordinate systems............... 78
4.2.2 Boundary conditions in the Cartesian coordinate systems .............................. 79
4.2.2.1 Left nodes.................................................................................................. 80
4.2.2.1.1 The line 0 ( ) x i = with prescribed temperature boundary condition... 80
4.2.2.1.2 The line 0 ( ) x i = with prescribed heat flux boundary condition........ 80
4.2.2.1.3 The line 0 ( ) x i = with convection boundary condition ...................... 81
4.2.2.2 Right nodes ............................................................................................... 81
4.2.2.2.1 The procedure on line ( ) x L iii = is similar with the left nodes......... 81
4.2.2.3 Bottom nodal lines .................................................................................... 82
4.2.2.3.1 The line 0 ( ) y ii = with prescribed temperature boundary condition . 82
4.2.2.3.2 The line 0 ( ) y ii = prescribed heat flux boundary condition .............. 82
4.2.2.3.3 The line 0 ( ) y ii = with convection boundary condition .................... 82
4.2.2.4 Upper nodal lines ...................................................................................... 83
4.2.2.4.1 The line ( ) y Hiv = with prescribed temperature boundary condition 83
4.2.2.4.2 The line ( ) y Hiv = with prescribed heat flux boundary condition .... 83
4.2.2.4.3 The line ( ) y Hiv = with convection boundary condition .................. 84
4.3 Proposed Method in the polar coordinate Systems................................................. 84
4.3.1 Finite difference discretization in the polar coordinate systems...................... 84
4.3.2 Boundary conditions in the polar coordinate systems ..................................... 85
4.3.2.1 Right nodes ............................................................................................... 86
4.3.2.1.1 The line 0 ( ) iii θ = with prescribed temperature boundary condition 86
4.3.2.1.2 The line 0 ( ) iii θ = with prescribed heat flux boundary condition ..... 86
4.3.2.1.3 The line 0 ( ) iii θ = with convection boundary condition ................... 86
4.3.2.2 Left nodes.................................................................................................. 87
4.3.2.2.1 The procedure on line / 2 ( ) i θ π = is similar with the left nodes ...... 87
4.3.2.3 Bottom nodal lines .................................................................................... 87
4.3.2.3.1 The arc
0
() rr ii = with prescribed temperature boundary condition.. 88
4.3.2.3.2 The arc
0
() rr ii = with prescribed heat flux boundary condition....... 88
4.3.2.3.3 The arc
0
() rr ii = with convection boundary condition ..................... 88
4.3.2.4 Upper nodal lines ...................................................................................... 89
4.3.2.4.1 The arc
0
() rr H iv = + with prescribed temperature boundary
condition ........................................................................................................... 89
4.3.2.4.2 The arc
0
() rr H iv = + with prescribed heat flux boundary condition89
4.3.2.4.3 The arc
0
() rr H iv = + with convection boundary condition ............. 89
4.4 Assembly of Governing Equations and Boundary Conditions............................... 90
4.5 State Space Formulations........................................................................................ 91
4.6 Solution by DTFM.................................................................................................. 93
v
4.7 Assembly of Multiple Rectangular Subsections..................................................... 93
4.8 Steady State Numerical Examples.......................................................................... 95
4.9 Unsteady State Numerical examples .................................................................... 105
4.10 Discussion........................................................................................................... 119

Chapter 5 Static and Dynamic Analysis of Plate Problems ...................................... 121
5.1 Problem Statement................................................................................................ 121
5.2 Proposed Method in the Cartesian coordinate Systems........................................ 121
5.2.1 Finite difference discretization in the Cartesian coordinate systems............. 121
5.2.2 Boundary conditions in the Cartesian coordinate systems ............................ 122
5.2.2.1 Left nodes................................................................................................ 123
5.2.2.1.1 The line 0 ( ) x i = with clamped boundary condition ........................ 123
5.2.2.1.2 The line 0 ( ) x i = with simple supported boundary condition .......... 123
5.2.2.1.3 The line 0 ( ) x i = with free edge boundary condition....................... 123
5.2.2.2 Right nodes ............................................................................................. 124
5.2.2.2.1 The procedure on line ( ) x L iii = is similar with the left nodes........ 124
5.2.2.3 Bottom nodal lines .................................................................................. 124
5.2.2.3.1 The line 0 ( ) y ii = with clamped boundary condition ...................... 125
5.2.2.3.2 The line 0 ( ) y ii = with simple supported boundary condition......... 125
5.2.2.3.3 The line 0 ( ) y ii = with free edge boundary condition ..................... 125
5.2.2.4 Upper nodal lines .................................................................................... 126
5.2.2.4.1 The line ( ) y Hiv = with clamped boundary condition .................... 126
5.2.2.4.2 The line ( ) y Hiv = with simple supported boundary condition....... 126
5.2.2.4.3 The line ( ) y Hiv = with free edge boundary condition ................... 127
5.3 Proposed Method in the polar coordinate Systems............................................... 127
5.3.1 Finite difference discretization in the polar coordinate systems.................... 127
5.3.2 Boundary conditions in the polar coordinate systems ................................... 128
5.3.2.1 Right nodes ............................................................................................. 129
5.3.2.1.1 The 0 ( ) iii θ = line with clamped boundary condition ..................... 129
5.3.2.1.2 The 0 ( ) iii θ = line with simple supported boundary condition........ 129
5.3.2.1.3 The 0 ( ) iii θ = with free edge boundary condition ........................... 129
5.3.2.2 Left nodes................................................................................................ 130
5.3.2.2.1 The procedures on / 2 ( ) i θ π = is similar with the left nodes......... 130
5.3.2.3 Bottom nodal lines .................................................................................. 130
5.3.2.3.1 The arc
0
() rr ii = with clamped boundary condition ....................... 131
5.3.2.3.2 The arc
0
() rr ii = with simple supported boundary condition ......... 131
5.3.2.3.3 The arc
0
() rr ii = with free edge boundary condition...................... 131
5.3.2.4 Upper nodal lines .................................................................................... 132
5.3.2.4.1 The arc
0
() rr H iv = + with clamped boundary condition ............... 132
5.3.2.4.2 The arc
0
() rr H iv = + with simple supported boundary condition . 132
5.3.2.4.3 The arc
0
() rr H iv = + with free edge boundary condition.............. 133
vi
vii
5.4 Assembly of Governing Equations and Boundary Conditions............................. 133
5.5 State Space Formulations...................................................................................... 135
5.6 Solution by DTFM................................................................................................ 137
5.7 Static Numerical Examples................................................................................... 137
5.8 Dynamic Numerical examples.............................................................................. 145
5.9 Discussion............................................................................................................. 149

Chapter 6 Conclusions.................................................................................................. 150

Bibliography…………………………………………………………………………… 152

Appendices……………………………………………………………………………... 161
Appendix A………………………………………………………………………….. 161
Appendix B………………………………………………………………………….. 174
Appendix C………………………………………………………………………….. 179

LIST OF TABLES
Table 3.1 displacements and stresses of the square region at ( 54
y
CCCT 0.5,0.5)
Table 3.2 displacements and stresses of the square region at (0.5, 0.5) 58
x
TCFC
Table 3.3 displacements and stresses of the T annular region at (1.5 CCC
θ
, / 3 π) 62
Table 3.4 displacements and stresses of the TC annular region at (1.5 FC
θ
, / 3 π) 66
Table 3.5 displacements and stresses of the square region at (1, 0.5) 68
x
CFFT FFF
Table 3.6 displacements and stresses of the square region at C 74
x
CFCT FF
Table 3.7 natural frequency of a 2-D elastic rectangular region 75 CCCC
Table 3.8 natural frequency of a 2-D elastic annular sector region 76 CCCC
Table 4.1 temperature of the TTTT square region at ( 96 0.5,0.5)
Table 4.2 temperature of the TH square region at ( 99 AC 0.5,0.5)
Table 4.3 temperature of the TTTT annular region at (1.5, / 3) π 101
Table 4.4 temperature of the TH annular region at AC (1.5, / 3) π 103
Table 4.5 temperature of the T L-shaped region at 104 AHHTHC (1 ,1)
Table 5.1 deflection of the CCCC rectangular plate at 139 (0.5,0.5)
Table 5.2 deflection of the CSCF rectangular plate at 141 (0.5,0.5)
Table 5.3 deflection of the CCCC annular sector plate at (1.5, / 3) π 142
Table 5.4 deflection of the CSCF annular sector plate at (1.5, / 3) π 144
Table 5.5 natural frequency of the rectangular plate 146 SSSS
Table 5.6 natural frequency of the CCCC rectangular plate 146
Table 5.7 natural frequency of the CSCF annular sector plate 147
Table 5.8 natural frequency of the CCCC annular sector plate 148
viii
LIST OF FIGURES
Figure 1.1 total issues in the dissertation 9
Figure 2.1 a rectangular continuum with mesh lines in the Cartesian coordinate systems 10
Figure 2.2 an elastic circular continuum with mesh lines in the polar coordinate systems 11
Figure 2.3 an L-shaped elastic continuum with mesh lines 12
Figure 2.4 an FDDTFM flow chart 29
Figure 3.1 an example of a rectangular continuum with boundary conditions 32
Figure 3.2 an example of a circular continuum with boundary conditions 39
Figure 3.3 an example of an L-shaped continuum with boundary conditions 49
Figure 3.4 a square region with boundary conditions 51
y
CCCT
Figure 3.5 distribution of displacement
x
δ on A-B for the square region 52
y
CCCT
Figure 3.6 distribution of displacement
y
δ on A-B for the square region 52
y
CCCT
Figure 3.7 distribution of stress
x
σ on A-B for the square region 53
y
CCCT
Figure 3.8 distribution of stress
y
σ on A-B for the square region 53
y
CCCT
Figure 3.9 distribution of shear stress
xy
τ on A-B for the square region 54
y
CCCT
Figure 3.10 a square region with boundary conditions 55
x
TCFC
Figure 3.11 distribution of displacement
x
δ on A-B for the square region 56
x
TCFC
Figure 3.12 distribution of displacement
y
δ on A-B for the square region 56
x
TCFC
Figure 3.13 distribution of stress
x
σ on A-B for the square region 57
x
TCFC
Figure 3.14 distribution of stress
y
σ on A-B for the square region 57
x
TCFC
Figure 3.15 distribution of stress
xy
τ on A-B for the square region 58
x
TCFC
Figure 3.16 an annular region with TC boundary conditions 59 CC
θ
ix
Figure 3.17 distribution of displacement
r
δ on A-B for the TC annular region 60 CC
θ
Figure 3.18 distribution of displacement
θ
δ on A-B for the TC annular region 60 CC
θ
Figure 3.19 distribution of stress
r
σ on A-B for the T annular region 61 CCC
θ
Figure 3.20 distribution of stress
θ
σ on A-B for the T annular region 61 CCC
θ
Figure 3.21 distribution of stress
r θ
τ on A-B for the TC annular region 62 CC
θ
Figure 3.22 an annular region with TC boundary conditions 63 FC
θ
Figure 3.23 distribution of displacement
r
δ on A-B for the TC annular region 64 FC
θ
Figure 3.24 distribution of displacement
θ
δ on A-B for the TC annular region 64 FC
θ
Figure 3.25 distribution of stress
r
σ on A-B for the TC annular region 65 FC
θ
Figure 3.26 distribution of stress
θ
σ on A-B for the TC annular region 65 FC
θ
Figure 3.27 distribution of stress
r θ
τ on A-B for the TC annular region 66 FC
θ
Figure 3.28 an L-shaped region with boundary conditions 67
x
CFFT FFF
Figure 3.29 distribution of displacement
u
δ on G-D for the L-shaped region 68
x
CFFT FFF
Figure 3.30 distribution of displacement
v
δ on G-D for the L-shaped region 69
x
CFFT FFF
Figure 3.31 distribution of stress
x
σ on G-D for the L-shaped region 69
x
CFFT FFF
Figure 3.32 distribution of stress
xy
τ on G-D for the L-shaped region 70
x
CFFT FFF
Figure 3.33 a square region with boundary conditions 71
x
CFCT FF
Figure 3.34 distribution of displacement
u
δ on A-B for the square region 72
x
CFCT FF
Figure 3.35 distribution of displacement
v
δ on A-B for the square region 72
x
CFCT FF
Figure 3.36 distribution of stress
x
σ on A-B for the square region 73
x
CFCT FF
x
Figure 3.37 distribution of stress
y
σ on D-E for the square region 73
x
CFCT FF
Figure 4.1 an example of rectangular heat conduction with boundary conditions 79
Figure 4.2 an example of circular heat conduction with boundary conditions 85
Figure 4.3 an example of L-shaped heat conduction with boundary conditions 94
Figure 4.4 the rectangular heat conduction with TTTT boundary conditions 96
Figure 4.5 distribution of temperature on A-B for the TTTT square region 97
Figure 4.6 the rectangular heat conduction with TH boundary conditions 98 AC
Figure 4.7 distribution of temperature on A-B for the TH square region 98 AC
Figure 4.8 the annular heat conduction with TTTT boundary conditions 100
Figure 4.9 distribution of temperature on A-B for theTTTT annular region 100
Figure 4.10 the annular heat conduction with TH boundary conditions 101 AC
Figure 4.11 distribution of temperature on A-B for the TH annular region 102 AC
Figure 4.12 the L-shaped heat conduction with T boundary conditions 103 AHHTHC
Figure 4.13 distribution of temperature on A-B for the T L-shaped region 104 AHHTHC
Figure 4.14 the Fourier series effect for example 4.6 110
Figure 4.15 the square heat conduction with TTFT boundary conditions 112
Figure 4.16 distribution of general temp. on A-B for the TTFT region by mesh lines 113
Figure 4.17 distribution of general temperature on A-B for the TTFT region by time 113
Figure 4.18 distribution of general flux on A-B for the TTF region by mesh lines 114 T
Figure 4.19 distribution of general flux on A-B for the TTF region by time 114 T
Figure 4.20 the composite heat conduction with TTTFTT boundary conditions 116
Figure 4.21 distribution of general temp. on A-B for theTTTFTT region by mesh lines 117
Figure 4.22 distribution of general temperature on A-B for theTTTF region by time 117 TT
Figure 4.23 distribution of general flux on A-B for the TTTFTT region by mesh lines 118
xi
Figure 4.24 distribution of general flux on A-B for the TTTFTT region by time 118
Figure 5.1 an example of rectangular plate with boundary conditions 122
Figure 5.2 an example of circular plate with boundary conditions 128
Figure 5.3 a square region plate with boundary conditions 138 CCCC
Figure 5.4 distribution of deflection on A-B for the CCCC rectangular region 139
Figure 5.5 a square region plate with boundary conditions 140 CSCF
Figure 5.6 distribution of deflection on A-B for the CS rectangular region 141 CF
Figure 5.7 an annular sector plate withCCCC boundary conditions 142
Figure 5.8 distribution of deflection on A-B for the CCCC annular sector plate 143
Figure 5.9 an annular sector plate withCSCF boundary conditions 143
Figure 5.10 distribution of deflection on A-B for the annular sector plate 144 CSCF

xii
ABSTRACT

This dissertation presents a novel developed semi-analytic mathematic scheme, the finite
difference distributed transfer function method (FDDTFM), succeeds both static/steady-
state and dynamic/unsteady-state analysis on the two-dimensional elasticity, heat
conduction, and plate problems in the Cartesian and the polar coordinate systems. The
scheme uses the finite difference method (FDM) along the direction and the
distributed transfer function method (DTFM) along the
/ yr
/ x θ direction. The FDDTFM
doesn’t need to re-derive equations because of changing boundary conditions; use series
function for complicated geometry.
The Navier’s displacement equations by the FDDTFM examine the static and dynamic
aspects of the linear two-dimensional elastic plane problems with arbitrary boundary
conditions in the two different coordinate systems. The static results of square, annular
sector, and L-shaped regions with arbitrary boundary conditions show up to 384 times
faster convergence than those of the FEM. The free vibration results of square and
annular sector regions show the similar phenomenon.
xiii
The dissertation applies the FDDTFM to solve the steady-state and unsteady-state heat
conduction problems on the two-dimensional region with arbitrary boundary conditions
in the two different coordinate systems. The steady-state results of square, annular sector
and L-shaped regions with arbitrary boundary conditions examples present up to 10 times
faster convergence by comparing with those of the FEM. Exact general time response
solutions show the one-dimensional heat conduction problems by the DTFM with the
residue theorem work very well. The general time response results of the two-
xiv
dimensional heat conduction on square and composite regions also present the
comparable fact.
The proposed method analyzes the static and dynamic parts of the Kirchhoff plate
problems with arbitrary boundary conditions in the two coordinate systems. The static
results of rectangular and annular sector plates with arbitrary boundary conditions are up
to 166 times faster convergence that those of the FEM. The free vibration results of
square and annular sector regions show the similar trend.
The FDDTFM guarantees the convergences of those two-dimensional elastic continua
problems. The appendices contain over 20,000 possibilities of the boundary conditions on
the simple geometry.

1
Chapter 1

Introduction

1.1 Background and Motivation
There always exists an alternative way to solve the two-dimensional elastic continua
problems. The dissertation develops an innovative method. Generally, it is difficult to
solve the two-dimensional elastic continua problems in simple way. The combined
method provides another thought on this kind of problems. The first method is the
analytical closed form distributed transfer function method (DTFM). The second method
is the numerical finite difference method (FD). The proposed finite difference distributed
transfer function method (FDDTFM) shows the investigation on the two-dimensional
elastic continua problems. Almost all the time, the analytical method and closed form
methods get accurate but tedious solutions. The analytical methods need professional
mathematical knowledge on the simple problems and those methods are difficult to solve
the real world problems. Since the finite difference method (FDM) utilizes in many
research fields, such as Economics, Psychology, Sociology and Engineering (Goldberg,
1986), it also applies on the elastic continua problems. The distributed transfer function
method (DTFM) is capable to solve one-dimensional elastic continua problems. The
analytic method is not easy to solve the two-dimensional elastic continua governing
equations. The dissertation wants to deal with the two-dimensional elastic problems, so it
is a quite important to look for the other useful mathematical technique. The author
2
chooses the finite difference method (FDM) to assist distributed transfer function method
(DTFM) because the finite difference technique is one of well-developed numerical
methods. The combination of fundamental theories, finite difference method, and
distributed transfer function method create a new mathematic algorithm called the
FDDTFM. This is a semi-analytic analysis for two-dimensional elastic continua problems.

1.2 Previous Research
In 1755, Euler wrote a book named Institutiones calculi differentialis, which starts to
discuss the calculus of finite differences. In 1841, Cauchy derived an equation for the
error from truncating finite difference interpolation series called Cauchy remainder term.
C. Runge was the first person who applied the finite difference method (FDM) into the
elasticity in 1908 (Timoshenko & Goodier, 1970). In 1922, Richardson used the FDM to
predict the weather in his book named Weather Prediction by Numerical Process. Crank
and Nicolson also studied a lot of FDM in 2nd order partial differential equations in 1947.
Flügge-Lotz was the first person who involved FDM in computer work in 1953. FDM
was a wide popular method in many fields, such as Economics, Psychology, Sociology
and Engineering (Goldberg, 1986). The choice of the FDM because it solved most
complicated linear and nonlinear partial differential equations with mixed boundary
conditions. FDM took fewer CPU time, fewer memory storage and hard driver space.
FDM had high accuracy on the computation on the partial differential equations. FDM
coped with the arbitrary shapes (Akanda, Ahmed, Khan, & Uddin, 2000), and mixed
boundary conditions very well (Ahmed, Khan, Islam, & Uddin, 1998). Ahmed, Idris, &
Uddin (1996) used the new function in terms of displacement components instead of Airy
3
stress function. They applied the finite difference method to solve the both end fixed deep
beam with the uniformly distributed loading on the top edge. This paper said the Airy
stress function had a difficulty on the mixed boundary conditions. Akanda, Ahmed, Khan,
& Uddin (2000) presented a new function in term of displacement components rather
than Airy stress function. This method applied the arbitrary shaped bodies with mixed
boundary conditions to solve the plan stress and plane strain problems. Finite element
method was too complicated and gets unclearly solution. The DQFDM was a modified
finite difference technique by Chen (2001). The method solved the complicated geometry
with irregular grids. The finite difference method avoided the too many meshes compared
with finite elements method. FDM didn’t need lots of computer works. Cocchi (2000)
presented the 3-dimensional elastic body static analysis by finite difference technique
with arbitrary grids. It managed the complicated structures with various loading and
boundary conditions. Dow, Jones, & Harwood (1990) provided a method to solve some
difficulties of finite difference method on elastic problem. The boundary conditions using
a set of constitutive equation represented the fictitious points. The higher accuracy in
displacement and stress and less computation works compared with the finite element
method. Ioakimidis (1998) used the existed experimental data to get the finite difference
theoretical equations by famous Buchberger algorithm for Grobner basis. This method
applied on the concentrated force, circular hole, and torsion problems. Savula, Mang,
Dyyak, & Pauk (2000) presented the hybrid numerical analysis on the two dimensional
elastic problem. This method called coupled boundary finite elements method. It used the
direct boundary element method for the solid with the theory of elasticity and finite
element method for the shell with the Timoshenko’s shell theory. Zeng & Liu (2001)
4
presented the heterogeneous complex Biot’s equation by finite difference method. The
paper showed the first-order, leap-frog, staggered-grid, finite difference method had a
higher accuracy than second-order, centered-grid finite difference method. Ahmed, Khan,
Islam, & Uddin (1998) used the function in terms of displacement combined with the
finite difference method to solve mixed boundary conditions deep beam problems. It
provided very well on critical region of cantilever beam, which no other methods beat it.
Du, Bancroft, and Dong (2004) used the finite element method combined with finite
difference method to solve the 2D wave equation. This method had FEM advantages such
as arbitrary domain, and loading boundary conditions. This method also had FD
advantages such as less memory storage and less CPU time for computation. This FE-
FDM provided the higher accuracy on this time relay partial differential equations.
Gavete, L., Gavete, M. L., & Benito (2003) use the generalized finite difference (GFD)
method which derived from the finite difference scheme to solve two-dimensional L
shaped elastic problem. It also compared with element free Galerkin (EFG) method. This
paper proved the GFD method was more accurate than EFG method even the GFD had
an ill-conditioned problems. They are all called meshless methods. The Stochastic finite
difference method (SFDM) was better than stochastic finite element method. SFDM
(Kaminski, 2001) was easier computation method than SFEM. SFEM needed the various
discretized nodal points defined or particular elements midpoints defined, but the SFDM
needed the network point defined only. Koh, Lee, & Itoh (1997) used the hybrid finite
difference and finite element method to solve the via hole problems. The finite element
method coped with the hole-region and the finite difference reduced the matrix size. This
hybrid method had both advantages. It also got the higher accuracy from its computation.
5
Biot was the first person who studied in heat conduction in 1802. Fourier developed
theory of heat conduction, boundary value problems, and differential equations in 1822.
In 1863, Grashof gave his famous lecture “theory of heat”. Reynolds studied the transfer
of heat between solids in 1875. Graetz is the first pioneer on the heat conduction,
radiation, friction and elasticity field before 1890. Prandtl discovered in 1904 of the
boundary layer in moving fluid. The well-known Prandtl number named after him. In
1910, Jakob started his outstanding career in thermodynamics and heat transfer. Nusselt
wrote “The Basic Laws of Heat Transfer” which he first proposed the dimensionless
groups in 1915. Schmidt was a pioneer in the Thermodynamics field, especially in Heat
and Mass Transfer in 1925. Colburn worked on heat and mass transfer along with
thermodynamic principles in 1940. de Monte (2003) applied separation of variables and
two 1-D eigenproblems to get the exact closed form solution. It has complicated
procedures. Chen & Chen, (1988); Chen, Lin, Wang, & Fang, (2002) combined the
numerical inverse Laplace transform method developed by Honig, G. & Hirdes, U. and
finite difference method to solve 2-D or 3-D transient heat conduction problems. Even
though they claimed it is an analytical solution, it is still an approximate method because
the way they choose coefficient while doing inverse Laplace transform. (Monde, M.,
Arima, H., Liu, W., Mitutake, Y., & Hammad, J. A., 2003)
In 1776, Euler was the first person who involved in plate free vibration analysis.
Bernoulli, J., Jr. made a modification on those experiences in 1789. Chladni used a thin
layer of sand to find out different modes of free vibration on plate in 1802. He also used
the strips Euler-D. Bernoulli beam theory to analyze the plate problems. In 1828,
Lagrange was the first person to provide the plate equation appropriately. Cauchy and
6
Poisson deduced the plate bending problems from elasticity theory around 1828. In 1823,
Navier got the plate thickness is function of flexural rigidity and he used the Fourier
series to get the exact solutions. Possion analyzed the circular plate’s free vibration in
1829. In 1850, Kichhoff gave the Kirchhoff’s hypotheses for plate bending theory. In
1893, Voight was the first person who gets the exact free vibration solution of rectangular
plate with two-opposite-sides simple supported. Levy solved most rectangular plates
which have two parallel simple supports and the other two arbitrary boundary conditions
in 1899. Timoshenko and Woinowsky-Krieger published a classic book related to plates
and shells in 1959. In 1909, Ritz extended the Rayleigh principle for getting vibration
frequencies. Gorman (1984) applied superposition method to get the natural frequency of
rectangular plates with mixed boundary conditions. The method is analytical but it is
complicated. Kocatürk, T., & Altinta ş, G. (2003) use the energy-based finite difference
method to solve viscoelastically point-supported plate problems. Even though it can
extend to various support and loading conditions, it still didn’t show results of the plate
with free edge. Biancolini, M. E., Brutti, C., & Reccia, L. (2005) used the Rayleigh
method to solve equation granted by Hearmon at first frequency of orthotropic plates.
The paper only showed the first frequency of plate, and didn’t show how to deal the
frequency of plate with free edge. Yang & Zhou (1997) use strip distributed transfer
function method based on Cheung’s finite strip method and Yang’s distributed transfer
function to get the deflection and natural frequency of annular plate.
The distributed transfer function method (DTFM) developed by Yang (1989; 1992; 1996)
is the most convenient tool for the one-dimensional distributed continua. The DTFM
deals with the symmetric and non-symmetric one-dimensional continua very well. The
7
DTFM can get the closed analytic form solution without solving the eigen-values and
eigen-vectors. (Yang & Tan, 1992; Yang & Zhou 1996)

1.3 Objectives and Scope of the Study
The dissertation represents the convergence and the investigation of the innovative
method, FDDTFM, in the two-dimensional elastic continua such as two-dimensional
elasticity, heat conduction and plate problems. Each problem involves in the Cartesian
and in the polar coordinate systems. The two-dimensional elasticity problems also work
on the L-shaped model. The two-dimensional heat conduction problems also work on the
composite material model. Each division discusses both the static/steady-state and
dynamic/unsteady-state field. The dissertation contains 15 issues totally. (Fig. 1.1) The
dissertation uses the novel FDDTFM to get the displacement, strain, stress, and natural
frequency in two-dimensional elasticity problems; the temperature and flux distribution
in two-dimensional heat conduction problems; the deflection, bending moment, shear
stresses, and natural frequency in plate problems. The chapter two mentions three major
models: rectangular, circular and L-shaped continua. The mesh lines, mesh points and
notation are introduced. The basic elasticity, heat conduction, plate and finite difference
schemes properties demonstrate secondly. The state variables are defined because the
finite difference method applied in this dissertation. The generalized distributed transfer
function method solution provides for two-dimensional elasticity, heat conduction and
plate problems. The chapter three shows the static and dynamics derivation on the two-
dimensional Navier’s governing equations with three major boundary conditions:
displacement, traction and mixed boundary conditions in the Cartesian coordinate
8
systems and the polar coordinate systems. The several static and dynamic numerical
examples and discussion show how the proposed method works. The chapter four
demonstrates each field in three major boundary conditions: prescribed temperature,
prescribed heat flux and convection boundary conditions. The dissertation shows the
unsteady state (transient) derivation on the one-dimensional heat conduction problems in
the Cartesian coordinate systems; the steady state derivation on the two-dimensional heat
conduction governing equations in the Cartesian coordinate systems and the polar
coordinate systems; the unsteady state (transient) derivation on the two-dimensional heat
conduction governing equations in the Cartesian coordinate systems. The several steady
state and unsteady state (transient) numerical examples and discussion show how the
proposed method succeeds. The chapter five explains the static and dynamics derivation
on the plate governing equations with three major boundary conditions: clamped, simple
support and free boundary conditions, in the Cartesian coordinate systems and the polar
coordinate systems. The several static and dynamic numerical examples and discussion
show how the proposed method happened as expected. The last chapter will conclude and
summarize the whole dissertation. The appendices A, B and C shows tons of derivation
results for the two-dimensional elasticity, heat conduction and plate problems in the
Cartesian and the polar coordinate systems. Those derivation procedures of the boundary
conditions with their corresponding governing equations are complicated, so the final
results make several tables in appendices.

Figure 1.1 total issues in the dissertation

9
Chapter 2

Basic Ideas

2.1 Problem Statements

Figure 2.1 a rectangular continuum with mesh lines in the Cartesian coordinate systems
Fig. 2.1 represents a rectangular elastic continuum in the Cartesian coordinate systems.
Its length is L and height is H. This elastic continuum along y axis separates into
mesh lines. The left line and right line contains 2 mesh points totally. The mesh points
are virtual mesh points. The proposed method presents a way to eliminate
those virtual mesh points. The mesh width along the
n
n
0 and 1 n +
y direction denotes . And this
rectangular elastic continuum has four edges which symbolize l along
h
eft edge ( ) i 0 x =
and 0 y H ≤≤ , bottom edge ( ) ii along 0 y = and 0 x L ≤ ≤ , along right edge ( ) iii x L =
and 0 along yH ≤≤ and upper edge ( ) iv yH = and 0 x L ≤ ≤ . The notation means
y
CT CF
10
that edge ( and ) i 0 x = 0 y H ≤≤ is clamped( . The edge ( ) C ) ii 0 y = and 0 x L ≤≤ has a
traction in ( ) T y direction. The outward direction is a positive sign; the inward direction is
a negative sign. The edge ( ) iii x L = and 0 y H ≤ ≤ is clamped ( . The edge ( ) C ) iv y H =
and 0 x L ≤≤ is free ( . ) F

Figure 2.2 an elastic circular continuum with mesh lines in the polar coordinate systems
Fig. 2.2 shows an elastic circular continuum in the polar coordinate systems. The
difference between inner radius and outer radius is . The inner circular circumference
is
H
00
r θ and outer circular circumference is
0
() rH
0
θ + . The radial coordinate from the pole
denotes . The angular coordinate from axis indicate O r
0
0 θ . The counterclockwise
direction is positive. This elastic continuum along r axis separated into n mesh lines.
The mesh width in the direction is . The left line and right line contains 2 mesh
points totally. The mesh points 0
r h n
and 1 n + are virtual mesh points. The present method
finds a way to eliminate those virtual mesh points. This circular elastic continuum has
four edges which are l alongrr eft edge ( ) i
0 0
r H ≤+ and
0
0
θ θ = , bottom edge ( ) ii along ≤
11
0
rr = and
00
0
0 θ θ ≤≤ , along right edge ( ) iii
00
rr r H ≤ ≤+ and
0
0 θ = ,
along and
and upper edge ( ) iv
0
rr H =+
0
0
0
0
θ θ ≤≤ . The notation means that edge (
and
FCT C
θ
) i
00
rr r H ≤≤ +
0
0
θ θ = is free( . The edge ( ) F ) ii
0
rr = and
0
0
0
0
θ θ ≤ ≤ is clamped . The edge
and has a traction in
( ) C ( ) iii
00
rr r H ≤≤ +
0
0 θ = () T θ direction. The outward direction has a
positive sign and the inward direction has a negative sign. The edge () iv
0
rr H = +
and
0
0
0
0
θ θ ≤≤ is clamped . ( ) C

Figure 2.3 an L-shaped elastic continuum with mesh lines
The L-shaped elastic continuum assembles from two rectangular substructures: Section I
(ABEDC) and Section II (CDGF). Section I with height and length has mesh
lines. The Section I contains 4 mesh points. Section II with height and length
has n mesh lines. The Section II contains 2 mesh points. The mesh width along
1 H 1 L 2n
n 2 H 2 L
n y axis
denotes . The nodes have two types: the boundary nodes and interconnected nodes. The
known boundary mesh points are (1 ~ ,
h
2 ini 1 ~2 nj nj + , and 1 ~ ). The unknown knk
12
interconnected mesh points are (1 ~ j nj ). The interconnected nodes have to consider the
displacement continuity and force balance. In heat conduction problems, the temperature
and heat flux at the contact surface have the same quantity. In the plate problems, the
displacement and rotational displacement have the same magnitude. (Park, 1996) The L-
shaped elastic continuum has seven edges which symbolize edge along () i 0 x =
and 0 1 y H ≤≤ , edge ( along ) ii 01 x L ≤ ≤ and 0 y = , edge along ( ) iii 112 L xL L ≤≤ +
and , edge ( along 0 y = ) iv 1 2 x LL = + and 0 2 y H ≤ ≤ , edge ( along ) v 1 L ≤ 12 x LL ≤ +
and 2 y H = , edge ( along ) vi 1 x L = and 2 1 Hy H ≤ ≤ , and edge ( along ) vii 0 ≤ 1 x L ≤
and 1 y H = . The notation means that the edge and
yx
FCFT FT F ( ) i 0 x = 0 1 y H ≤≤ is
free . The edge ( ( ) F ) ii01 x L ≤ ≤ and 0 y = is clamped . The edge
and is free . The edge (
( ) C ( ) iii
11 Lx L L ≤≤ +2 2 0 y = ( ) F ) iv 1 x LL = + and 0 2 y H ≤≤ has a
traction T in y direction. The edge ( ) v11 Lx L L2 ≤ ≤+ and 2 y H = is free . The edge ( ) F
() vi 1 x L = and has a traction T in 2 1 Hy H ≤≤ x direction. The edge( ) vii01 x L ≤≤ and
y = 1 H is free . ( ) F

2.2 Background Theories
The whole dissertation investigates the two-dimensional elastic continua problems, such
as two-dimensional elasticity, two-dimensional heat conduction, and plate problems.
2.2.1 Elasticity
Truss and beam are slender elements. Their length is much greater than their cross
section area. Truss supports only tension or compression along its length. Beam supports
lateral loads which cause flexural bending. Torsion has the same properties as truss but
13
additionally supports rotation. Two-Dimensional solid is an element which applies
external loads on the same plane. Plate is an element with external loads acting out of the
plane. The out of plane loads cause flexural bending. Shell is an element similar to a plate
but typically shell uses on curved surface and supports both in plane and out of plane
loads. Three-Dimensional solid is an element that obeys the strain displacement and
stress strain relationships.
Elasticity defines the material behavior will return its original state when the load
removes. The isotropic property defines the material behavior under stress is the same as
in all directions. The homogenous property defines the material behavior is the same as at
every point in a body. If the material is homogeneous, it should be continuous too.
Generally, the elasticity theory deals with the three-dimensional problems. Under plane
stress or plane strain assumption, the three-dimensional problems reduce to two-
dimensional problems. The ratio of the thickness of the plane to the length of the plane is
pretty small, so the plane stress assumption applies. (i.e., 0
zxz yz
σ ττ = == ) The ratio of
the thickness of the plate to the length of the plane is pretty large, so the plane strain
assumption applies. (i.e., 0
zxz yz
ε γγ = == ) The plane stress assumption demonstrates
throughout whole dissertation. The equilibrium equations in the Cartesian coordinate
systems are:
0; 0
xy xy yy
x
xy
ff
xy x y
ττσ
σ
∂∂∂
∂
+ += + + =
∂∂ ∂ ∂
(2.1)
The normal stress on the surface represents i
i
σ . The j direction shear stress oni surface
means
ij
τ . The body forces on the surface shows i
i
f . The body force could be, magnetic
14
and inertia force denotes
i
f . The inertia forces are
22
(/ ut ρ ) − ∂∂ and . The
equilibrium equations show a body in equilibrium state. The Hooke’s equation shows the
relations between strains
22
(/ vt ρ −∂ ∂)
,,
x yxy
ε εγ and stresses ,,
x yxy
σ στ followed the linear
homogeneous, continuous, and isotropic elastic hypotheses. The modulus of elasticity in
tension is . The Possion’s ratio denotes . E v

2
10
10
1
1
00
2
xx
y
xy xy
v
E
v
v
σε
σ
ν
τγ
⎡⎤
⎢⎥ ⎡⎤ ⎡ ⎤
⎢⎥ ⎢⎥ ⎢ ⎥
=
⎢⎥ ⎢⎥ ⎢ ⎥
−
⎢⎥ ⎢⎥ ⎢ ⎥
−
⎣⎦ ⎣ ⎦
⎢⎥
⎣⎦

y
ε (2.2)
The strain-displacement relation equation represents a link between displacement and
strain.
; ;
xy xy
uv uv
x yy
εε γ
x
∂ ∂∂
== =+
∂
∂ ∂∂∂
(2.3)
They are derived from geometry calculus.
Now, strain-displacement relation equations (2.3) and Hooke’s equations (2.2) substitute
into equilibrium equations (2.1), The Navier’s equations in the Cartesian coordinate
system are:

22 2
22 2 2
22 2
222 2
0
12(1) 1 2(1)
0
2(1 ) 1 1 2(1 )
x
y
Eu E u vE E v
f
vx v y v v xy
Ev E v vE E u
f
vx v y v v xy
⎛⎞ ∂∂ ∂
++ + +
⎜⎟
−∂ + ∂ − + ∂∂
⎝⎠
⎛⎞ ∂∂ ∂
++ + +
⎜⎟
+∂ − ∂ − + ∂∂
⎝⎠
=
=
n
(2.4)
2.2.2 Heat conduction in two-dimensional region
The law of heat conduction for isotropic body represents /
flux
qkT =−∂ ∂ . This law of
heat conduction is the same as Fourier law of heat conduction. The heat flux is
flux
q . The
15
thermal conductivity of the solid is . The direction denotes . The temperature
corresponds toT .
k n
The energy-in is . The energy-out stands for
i
Q
idi
Q
+
. The energy generated within element
indicates
gen
Q . The change in internal energy stored is dE dt .

12 112 2
nn gen ndnndn
dE
QQ Q Q Q
d τ
++
++ = + + (2.5)

[()
[()]
TT T T
kdy kdx gdxdy k k dx dy
xy xx x
TT T
k k dy dx cdxdy
yy y t
ρ
∂∂ ∂ ∂∂
−− + =− +
∂∂ ∂∂ ∂
∂∂ ∂ ∂
−+ +
∂∂ ∂ ∂
]
(2.6)
The general heat conduction equations are from the conservation of energy on a small
two-dimensional element,

22
22
TT g cT
x yk k t
ρ ∂∂
++ =
∂
∂∂∂
(2.7)
The energy generated per unit volume is . The density of the body presents g ρ .The
specific heat of material denotes . The thermal diffusivity of the material
defines
c
k
α =
c ρ
. The two-dimensional heat conduction in steady state with no heat
generation considers first.
2.2.3 Plate
The plate is elastic, isotropic and homogeneous material basically. It is flat in the
beginning. The mid-plane deflection is very small, so is this deflection slope. The vertical
line normal to the mid-plane maintain straight all the time. The shear forces and
rotational inertia are ignored. Those assumptions call the Kirchhoff-Love hypotheses.
16
The Kirchhoff-Love plate equations come from the constitutive, kinematics, resultant,
and equilibrium equations. The constitutive equations from the Hooke’s law are in
equation (2.2). The Kinematics equations show the connection between strain and
curvature.

22 2
22
2
2
T
xx
yy
xy xy
k
ww w
zk z
x yxy
k
ε
ε
γ
⎡⎤ ⎡ ⎤
⎡ ⎤ ∂∂ ∂ ⎢⎥ ⎢ ⎥
==−
⎢ ⎥
⎢⎥ ⎢ ⎥
∂∂ ∂∂
⎣ ⎦
⎢⎥ ⎢ ⎥
⎣⎦ ⎣ ⎦
(2.8)
The curvature of the deflected middle surface is .The distance from the unbending
middle surface shows . The displacement of plate denotes .
ij
k
z w
The bending moments ,
x y
M M and twisting bending moments
xy
M are integrating stress
resultants with thickness.

/2
/2
;
t
t
xx
h
y
h
xy xy
M
y
M zd
M
σ
σ
τ
−
z
⎡⎤⎡⎤
⎢⎥⎢⎥
=
⎢⎥⎢⎥
⎢⎥⎢⎥
⎣⎦⎣⎦
∫
(2.9)
The shear forces , and
xz
Q
yz
Q are integrating shear stresses.

/2
/2
;
t
t
h
xz xz
h
yz yz
Q
dz
Q
τ
τ
−
⎡⎤⎡⎤
=
⎢⎥⎢⎥
⎣⎦⎣⎦
∫
(2.10)
The moment summation about the x axis,
xy
x
xz
M
M
Q
x y
∂
∂
=+
∂ ∂
(2.11)
The moment summation about the y axis,
xy
yz
y
M M
Q
x y
∂ ∂
=+
∂ ∂
(2.12)
The force summation in the axis, z
yz
xz
z
Q
Q
p
xy
∂
∂
+ =−
∂∂
(2.13)
17
Substituting the constitutive, kinematics, and stress resultant, into equilibrium equations
get the plate governing equation.

3 44 4 2
422 4 2
(2 ) ;
12(1 )
t
tz
Eh ww w w
Dhp
2
D
x xy y t v
ρ
⎡⎤ ∂∂ ∂ ∂
++ + = =
⎢⎥
∂∂∂ ∂ ∂ −
⎣⎦
(2.14)
The constant flexural rigidity of homogeneous isotropic plate is . This governing
equation for thin plate is under Kirchhoff-Love plate assumptions. The Kirchhoff-Love
plate discusses throughout this dissertation. The modulus of elasticity in tension is . The
Possion’s ratio denotes . The plate thickness is . The vertical distributed load of
intensity is
D
E
v
t
h
z
p . The mass density of the material is ρ .

2.3 Boundary Conditions
2.3.1 Two-dimensional elasticity problems
There are three kinds of boundary conditions in two-dimensional elasticity problems:
displacement boundary conditions, traction boundary conditions and mixed boundary
conditions.
2.3.1.1 Displacement boundary conditions

00 0
; uuv v
0
= = . (2.15)
The initial displacements are
0
, and u
0
v . If the case is fixed boundary condition, it means
the initial displacement is zero.
2.3.1.2 Traction boundary conditions

xlxym
xyl y m
Ue
Ve
στ
τσ
=+
=+
e
e
G G G G (2.16)
18
19
m
e
G The cosine direction of unit outward normal to the boundary denotes . These
symbols, and ,satisfy in the Cartesian coordinate systems or the polar coordinate
systems. The components of surface forces per unit area, traction, on the boundary
are
and
l
e
G l
e
G m
e
G , and U V . The stresses and shear stresses are , and σ τ .
2.3.1.3 Mixed boundary conditions
The boundary conditions combined displacement boundary conditions with traction
boundary conditions.
2.3.2 Heat conduction in two-dimensional region
There are many kinds of heart conduction boundary conditions in two-dimensional region:
prescribed surface temperature, adiabatic surface temperature, prescribed heat flux
through the boundary surface, convection, and radiation.
2.3.2.1 Prescribed surface temperature
(,)(,) TPt Pt = Θ (2.17)
The position point on the surface is . The time denotes . The prescribed function of
position and time represents .
P t
( , ) Pt Θ
2.3.2.2 Prescribed heat flux through the boundary surface

(,)
(,)
flux
TPt
kq
n
Pt
∂
−=
∂
; (2.18)
The outward normal to the surface denotes . The differentiation along the normal
direction represents
n
n
n
∂
∂
. Perfectly adiabatic means no heat flux crosses the boundary.
(i.e., ) ( , ) / 0 TPt n ∂∂=

2.3.2.3 Heat transfer by convection and radiation

44
() (
gen c
T
ghT FT
n
λσ
∂
−+ = −Θ+ −Θ
∂
) (2.19a)
() , if
gen c r
T
ghT hh h T
n
λ
∂
−+ = −Θ = + −Θ<
∂
< (2.19b)
The summation of heat conduction and heat generation are equal to heat convection on
the boundary surface. (i.e., (
gen c
T
ghT
n
λ )
∂
−+ = −Θ
∂
). The summation of heat conduction
and heat generation are equal to heat radiation on the boundary surface. (i.e.,
gen
T
g
n
λ
∂
−+
∂

).
3
4( ) ( ) if
r
FT hT T σ ≅Θ −Θ= −Θ −Θ<<
2.3.2.4 Contact between two solids

12
1
12
TT
T
kk
nn
2
T
=
∂ ∂
=
∂ ∂
(2.20)
The two solid bodies have perfect thermal contact, so the temperature and heat flux at the
contact surface have the same quantity.

2.3.3 Plate problems
2.3.3.1 Clamped side
The deflection and angle of the plate are zero. The deflection of the plate denotes w . The
angle of the plate denotes θ .

0
0
x
x
w
w
x
θ
=
∂
= =
∂
(2.21)
20

2.3.3.2 Simple support side
The deflection and the bending moment of the plate are zero along the edge.

22
22
0, and ( ) 0
xx x
ww
wMD v
xy
∂∂
==− +
∂∂
x
= (2.22)
The term
22
0
x
wy ∂∂ = because is function of w x only.
2.3.3.3 Free side
Poisson (1829) formulated three boundary conditions: bending moment
x
M , twisting
moment
xy
M and shear force
x
Q are equal to zero. Kirchhoff (1850) proposed an effective
shear force per unit length
xy
xx
M
VQ
y
∂
=+
∂
.

22
22
[]
xx x
ww
MD
xy
ν
∂∂
0 =−+
∂∂
=
22
22
33 3
32
[]
2[(1 ) ] [ (2 ) ]0
x xxy x xy xy x
xx x
ww
V Q My M x My My D v
xx y
ww w
DD
yxy x xy
νν
∂∂ ∂
= +∂∂=∂ ∂+∂∂+∂ ∂=− +
∂∂ ∂
∂∂ ∂ ∂
−− =−+− =
∂∂∂ ∂ ∂∂
x
(2.23)

2.4 Several Important Equations in the polar coordinate Systems
The dissertation discusses the two-dimensional elastic continua in the polar coordinate
system. The equilibrium equations in the polar coordinate systems are:

1
0
1
20
rr r rr
r
rr
f
rr r
f
xr r
θθθ
θθθ θ
θ
σσ σ σ
θ
σσ σ
θ
∂∂ −
++ +=
∂∂
∂∂
++ +=
∂∂
(2.24)

21
The strain-displacement relation equations in the polar coordinate systems are:

11
; ;
rr r
uu v uv
rrr r r
θθ θ
εε γ
θθ
∂∂ ∂
==+ = +
∂∂ ∂∂
v
r
∂
− (2.25)
Strain-displacement relation equations (2.25) and Hooke’s equations (2.2) substitute into
equilibrium equations (2.24), the Navier’s equations in the polar coordinate system are:

22 2
22 2 2 2
22 2 2
11
1 2(1 ) 2(1 ) 1
(3 ) 1
0
2(1 ) 1
r
Eu v u E
rrr r
vE v E u
f
rr
νν θ νθ ν
νθ ν
∂Ε ∂ Ε ∂
++ +
−∂ − ∂∂ + ∂ − ∂
−∂
−− +=
−∂ −
1u
rr
∂
(2.26a)

22 2
2222
22 2
11
2( 1 ) 2( 1 ) 1 2( 1 )
(3 ) 1
0
2( 1 ) 2( 1 )
vuEv
rrr r
vE u v
f
rr
θ
ννθ νθ ν
νθ ν
Ε∂ Ε ∂ ∂ Ε ∂
++ +
+∂ − ∂∂ − ∂ + ∂
−∂ Ε
+− +=
−∂ +
1v
rr
(2.26b)
The relationships between the Cartesian coordinate systems and the polar coordinate
systems derive from geometrical relation cos xr θ = ,sin yr θ = and chain rules
ffr f
x rx x
θ
θ
∂∂∂ ∂∂
=+
∂∂∂ ∂ ∂
&
ffr f
y ry y
θ
θ
∂∂∂ ∂∂
=+
∂∂∂ ∂ ∂
.

22 2 2 2
2
22 2 2
22 2 2 2
2
22 2 2
22 2
22
sin 2 sin sin 2 sin
cos
sin 2 cos sin 2 cos
sin
sin2 cos2 sin2 cos2 sin
22
2
2
2
2
f ff f f
xr rr rrr r
f
f ff f f
yr rr rrr r
ff f f f
xy r r r r r r
θθ θ θ
θ
θθ
θθ θ θ
θ
θθ
θθ θ θ
θθ
∂∂ ∂ ∂ ∂ ∂
=− + + +
∂∂ ∂∂ ∂ ∂ ∂
∂∂ ∂ ∂ ∂ ∂
=+ + − +
∂∂ ∂∂ ∂ ∂ ∂
∂∂ ∂ ∂ ∂
=+ − + −
∂∂ ∂ ∂ ∂ ∂ ∂
f
θ
θ
2
22
2
2
f
r
θ
θ
∂
∂
(2.27)
The two-dimensional heat conduction equation in the polar coordinate systems is:

22
222
11 TT Tg c
rrr r k k t
ρ
θ
∂∂ ∂
++ +=
∂∂ ∂
T∂
∂
(2.28)

22
The plate equation in the polar coordinate systems is:

43 2 4 3 2
4 3 2 2 3 2 2 2 3 2 42 44
2
2
21 1 2 2 4 1
t
ww w w w w w
rrr r r r r r r r r r r
hw p
Dt D
4
w
θ θθ
ρ
∂∂ ∂ ∂ ∂ ∂ ∂ ∂
+− + + − + +
∂ ∂ ∂ ∂ ∂∂ ∂∂∂∂
∂
+=
∂
θ
(2.29)
In the polar coordinate systems, the boundary conditions of plate are more complicated.
The radial moment is
22
22
11
[(
r
ww
MD v
rrrr
2
)]
w
θ
∂∂ ∂
=− + +
∂ ∂∂
. (2.30)
The tangential moment is
22
22
11
[
ww
MDv
rrr r
θ
2
]
w
θ
∂∂∂
=− + +
∂ ∂∂
. (2.31)
The effective shear force with outward normal in the r direction is

22 2
222 2
11 1 1 1
[( ) ( )
r
ww w v w w
VD
rr r r r r r r r
]
θ θθθ
∂∂∂∂ −∂ ∂ ∂
=− + + + −
∂∂ ∂ ∂ ∂ ∂∂ ∂
. (2.32)
The effective shear force with outward normal in the θ direction is

22
222
)
w 11 1
[(
ww
VD
rr rrr
θ
θ θ
∂∂ ∂ ∂
=− + +
∂∂ ∂ ∂
2
2
11
(1 ) ( )]
ww
v
rr r r θ θ
∂∂ ∂
+− −
∂ ∂∂ ∂
. (2.33)

2.5 Finite Difference Schemes
Finite difference method is one of numerical methods to transform the governing
equations (ordinary derivative equations or partial derivative equations) and boundary
conditions into the algebra equations. There are several steps for the finite difference
scheme. 1. Discrete the objective into several finite mesh lines. 2. Use the forward,
backward, or central finite difference schemes. 3. Apply the boundary conditions for each
mesh node. 4. Solve the whole problem.
23
In two-dimensional elastic problems, the partial derivative of a function f depends on
two variables. There are many finite difference schemes. The forward difference scheme
is
1 kk k
f
ii
ff
nn
+
∂−
≈
∂Δ
1, 2. i = . The backward difference scheme is
1 kk k
ii
f ff
nn
−
∂ −
≈
∂Δ
. The Central
finite difference scheme is
1
2
kk k
ii
1
f ff
nn
+ −
∂−
≈
∂Δ
. The two variables and either represent
1
n
2
n
, x y in the Cartesian coordinate systems or , r θ in the polar coordinate systems. The high
order finite difference schemes in this dissertation derived from the central finite
difference schemes. The 2
nd
order finite difference scheme is
2
11
22
2
kk k k
ii
f ff f
nn
+−
∂−+
≈
∂Δ
.The
3
rd
order finite difference scheme is
()
3
21 1
3 3
22
2
kk k k k
i
i
2
f ff f f
n
n
+ +−
∂− + +
≈
∂
Δ
−
. The 4
th
order finite
difference scheme is
()
4
21 1
4 4
46 4
kk k k k k
i
i
2
f ff f f f
n
n
+ +−
∂− +− +
≈
∂
Δ
−
.

2.6 State Space Formulation
What the new matrix form governing equations as following are substitute boundary
conditions into two-dimensional governing equations by finite difference scheme along
y axis.

() () () 1
;, , ; , , ;
jj j
AU BU CU P A B C U P
× × ″′
++ =∑+ ∈ ∑ ∈ \\
2 for elasticity problems; for heat conduction and plate problems jn j n == (2.34)
24
In the two-dimensional elasticity problems, the symbol ∑ is
22
(1 )
k
k
kx
ky
uf
vh
vf E
ρ
ρ
−+ ⎡⎤
−
−
⎢⎥
−+
⎢⎥
⎣⎦

for
the Cartesian coordinate systems and is
22
h
2
(1 )
k
k
kx
ky
uf
vr
vf E
ρ
ρ
−+ ⎡ ⎤
−
−
⎢ ⎥
−+
⎢ ⎥
⎣ ⎦

for the polar
coordinate systems. In the two-dimensional heat conduction problems, the
symbol is ∑
2
k
ch
T
k
ρ
−
for the Cartesian coordinate systems and is
22
2
k
cr h
T
k
ρ
−
for the
polar coordinate systems. In the plate problems, the symbol ∑ is
4
t
k
hh
w
D
ρ
− for the
Cartesian coordinate systems and is
44
2
t
k
hr h
w
D
ρ
− for the polar coordinate systems.
The state variables vectors define as following,
[ ]
12 3 4 1
(,) ( ,) (,) ( ,) (,) ( ,)
T
nn
Ugt gt gt gt gt gt
−
=Δ Δ Δ Δ Δ Δ " . (2.35)
The displacement pairs , temperatureT , or deflection of the plate denotes as & uv w
i
Δ .
Rearrange the state variables vectors by

[ ]
[]
for elasticity and heat conduction problems
for plate problems
T
T
UU
UU U U
η
η
′ =
′ ′′ ′′′ =
. (2.36)
The equation (2.34) becomes the first order ordinary differential equation.
(, ) ( ) (, ) (, ); (0, ), or
d
gt F t gt q g t g m m L
dg
ηη =+ ∈ ∈Θ (2.37)
The matrices F and will state very detail in the following chapters. Since the top and
bottom boundary conditions substitute into the two-dimensional governing equations,
those procedures construct a new first order ordinary differential equation. The two-
q
25
dimensional elasticity problems have four-side boundary conditions. Now it is about to
deal with the other two boundary conditions, left-side and right-side.
()(0,)()(,)() Mtt Nt mt Rt η η + = (2.38)

2.7 Solution in Frequency Domain by Distributed Transfer Function
Method
The governing equations (2.37) and the inhomogeneous boundary conditions (2.38)
convert from time domain to s -domain (i.e., the frequency domain) by taking Laplace
transform. The use of the sate space variables obtains the matrix form two-dimensional
governing equations. The new first order matrix form ordinary differential equations are
obtained because of redefined state space variables. Applying the Green function solves
the new equations. The computation method is a closed form, and semi-analytic solution.
It doesn’t need to solve the eigenvectors and eigenvalues. The frequency response
function decides the stability of complex systems. Those governing equations have one
and only one solution, which means well-posed.

0
()
()( )
() () 1
(, ) (, , ) ( , ) (, ) ( ), (0, )
(, ) ( ) ,
(, , )
(, ) ( ) ,
(, ) ( ( ) ( ) )
m
Fs
Fs m
Fs g Fs m
gsGgsqsd HgsRs g
Hg s M se g
Gg s
Hg s N se g
Hg s e Ms N se
ξ
ξ
ηξξξ
ξ
ξ
ξ
−
−
−
=+
⎧ ≤
=
⎨
−>
⎩
=+
∫
m∈
(2.39)
For the Cartesian coordinate systems, gx = andmL = . For the polar coordinate systems,
g θ = and . m =Θ

26
2.8 Solution in Time Domain by Distributed Transfer Function Method
In the transient analysis, the solution depends on time. The dissertation provides a semi-
analytical method based on inverse Laplace transform for getting time domain solution.
This section introduces how a new procedure makes the time domain solution succeed.
Continuing the equation (2.39),

() ( )( )
0
( , ) ( , ) () ( , ) ( , ) () ( , )
( , ) ( ), (0, )
gm
Fs F s m
g
gs H g s M s e q s d H gs N s e q s d
Hg s R s g m
ξξ
η ξξ ξξ
−−
=−
+∈
∫∫
(2.40)

() () 1
() ()( )
0
() 1
( , ) ( () () )
[ ( ) ( , ) ( ) ( , ) ( )]
( ) ( , )
Fs g F s m
gm
Fs Fs m
g
Fs g
gs e M s N s e
Ms e qsd Ns e qsd Rs
eZ sWgs
ξξ
η
ξξ ξξ
−
−−
−
=+
×−
=
∫∫
+ (2.41)
For simplifying equations, let () Z s is equal to and is equivalent
to
()
() ()
Fs m
Ms N se + (, ) Wg s
() ( )( )
0
() ( , ) ( ) ( , ) ()
gm
Fs F s m
g
Ms e qsd Ns e qsd Rs
ξξ
ξξ ξξ
−−
−
∫∫
+ . In order to get the time
domain solution, the convolution integral and residue theorem apply in the governing
equations. According to the paper written by Yang, B., & Wu, X (1998), the residue
1
Res ( )
n
s
Z s
λ
−
=
is
()
T
nn
T
nn
uv
vZ u λ ′
n
. The right eigenvectors are and the left eigenvectors are .
(i.e.,
n
u
n
v
( ) 0
nn
Zu λ = and () 0
T
nn
Zv λ = ).
The matrices M and are independent of time throughout the dissertation, so N () Z s ′ is
equal to
()
n
Fm
n
de
N
d
λ
λ
.To solve
()
n
Fm
n
de
d
λ
λ
, and the first Gateaux derivative introduced.

() ( ) ( )
0
lim
2
nn
Fm F m F m
n
de e e
J
d
λλλ λ
λ
λ
+Δ − Δ
Δ→
−
==
Δ
n
λ
λ
(2.42)
27
Finally, the time domain solutions by distributed transfer function are following,

() () 1
0
1
() ( )
0
1
() ( )
0
1
(,) Res[ ( )] (, )
( , )
( , ) .
n
n
nn
nn
t
t Fs g
n
s
n
T
t
tFg nn
n T
n
nn
T
t
Fg t nn
n T
n
nn
gte eZsWg
uv
ee Wgd
vNJu
uv
eWge
vNJu
λτ
λ
λτ λ
λλ
d
d
τ
η λτ
λ τ
λ τ
∞
−− −
=
=
∞
−−
=
∞
−−
=
=
=
=
∑
∫
∑
∫
∑
∫
(2.43)
2.9 Flow Chart
In the two-dimensional elasticity problems, the equilibrium equation comes from basic
free body diagram. The Hooke’s law and the strain-displacement relationship obtain the
two dimensional elastic Navier governing equations. In the two-dimensional heat
conduction problems, the energy in conducted and heat generated within element equal to
energy out conducted and change in internal energy. The heat conduction governing
equations in two-dimensional region are from the conservation of energy. In the plate
problem, substituting the constitutive, kinematics, and stress resultant, into equilibrium
equations get the plate governing plate equations based on the Kirchhoff-Love hypothesis.
Once the governing equations are granted, the finite difference methods apply along
y or axis. The boundary conditions on the edge for two-dimensional elasticity, heat
conduction in two-dimensional region, or plate problems substitute into the governing
equations. Three major boundary conditions demonstrate for each two-dimensional
elastic continuum problems. The state space formulation constructs the matrix form
governing equations and matrix form boundary conditions. Those two-dimensional
governing equations become , matrices and boundary conditions turn out
to be
r
, AB and C P
, and MN R matrices. By rearranging the state space variables, the , & , AB C
28
P matrices become F & matrices. After all matrices have derived, the distributed
transfer function method applies on these two-dimensional elastic continua problems.
Finally, the proposed method (FDDTFM) gets the static/steady-state or dynamic/transient
solutions along
q
x or θ direction. The FDDTFM obtains placement, stress and strain in the
two-dimensional elasticity problems; finds the temperature and flux in the two-
dimensional heat conduction problems; gain the deflection, slope, bending moment, and
shear force in the plate problems.

Figure 2.4 an FDDTFM flow chart

29

Chapter 3

Static and Dynamic Analysis of Two-dimensional Elasticity Problems

3.1 Problem Statement
If the behavior of elastic continua is under elastic range, the three-dimensional linear
elastic problems can reduce to two-dimensional elastic problems. This dissertation
provides a new method called the finite difference distributed transfer function method
(FDDTFM) to solve the two-dimensional linear elasticity problems. The equilibrium
equation (2.1) in term of displacement, Navier equations in the Cartesian coordinate
systems (2.4), can be expressed by substituting generalized Hooke’s Law (2.2) and strain-
displacement equations (2.3). The equilibrium equation (2.24) in term of displacement,
Navier equations in the polar coordinate systems (2.26), can be expressed by substituting
generalized Hooke’s Law (2.2) and strain-displacement equations (2.25). The proposed
method uses three major boundary conditions (2.15; 2.16) to solve two-dimensional
rectangular, circular, and L-shaped elastic continua problems.

3.2 Proposed Method in the Cartesian coordinate Systems
3.2.1 Finite difference discretization in the Cartesian coordinate systems
The current case contains n mesh points; mesh lines (Fig. 2.1). The displacement on
mesh line located at
n
th
k (, ) x kh defines . The subdivision height between two mesh
k
u
30
points is h . The total height of the elastic continuum is . The total width of the elastic
continuum is .
H
L
(, ) (, ); (, );
kk
H
uuxky uxkh v vxkh y h
k
=Δ= = Δ== (3.1)
The displacement on the x direction is . The displacement on the
k
u y direction is . The
Finite difference method along direction applies on the Navier’s governing equations
in the Cartesian coordinate systems (2.4).
k
v
y
The Central finite difference scheme for Navier’s equations in the Cartesian coordinate
systems is introduced.

11
11
22 2
2
2
11
11
22
2
2
1
(2 )( )(
12(1) 1 2(1)2
0
1
()( ) (2
12(1) 2 2(1) 1
0
kk
kkkk
k
xk
kk
kkk
k
yk
EE vE Ev
uuuu
vh v v v h
u
f
t
vE E u u E E
vvv
vv h v hv
v
f
t
ρ
ρ
+−
+−
+−
+−
′′ −
′′+−+++
−+ − +
∂
−+ =
∂
′′ −
′′ ++ + −
−+ + −
∂
−+ =
∂
2
)
)
k
v
v+
(3.2)
3.2.2 Boundary conditions in the Cartesian coordinate systems
In this dissertation, there are four-side boundary conditions for two-dimensional elasticity
problems.
The proposed method concerns the OB & the AD sides first in Fig. 3.1. After that, it
regards as the other two opposite sides, the OA& the BD sides. The notation
means that edge is traction ( with
x
TCFC ( ) i ) T x direction on the edge. The outward
direction is positive sign. The inward direction is negative. The edge is clamped .
The edge is free . The edge is clamped . The detail information of the
() ii () C
() iii () F () iv () C
31
mesh lines and the Cartesian coordinate component is in Figure 2.1. The dissertation is
going to investigate the middle dotted line . ( ) m ""

Figure 3.1 an example of a rectangular continuum with boundary conditions.
The first part considerations want to make form such as () (0, ) () ( , ) ( ) Mtt Nt Lt Rt η η += .

11
11
(,) [ (,), (,) ( ,), ( ,),
( , ), ( , ) ( , ), ( , )]; 0 or
nn
nn
mt u mt v mt u mt v mt
u mtv mt u mtv mt m L
η =
′′ ′ ′ =
" " (3.3)
3.2.2.1 Left nodes
3.2.2.1.1 The line 0 ( ) x i = with traction boundary condition
This current case uses the central finite difference scheme throughout the whole mesh
points except the boundary mesh points, and use forward and backward finite difference
scheme on the term. This technique avoids the extra unknown terms. 1 and k = n
The central finite difference scheme applies to the traction boundary conditions,

11
22 2 2
11
;
111 1 2
2(1 ) 2(1 ) 2(1 ) 2 2(1 )
k
k
xxlxy yl
kk
k
kk
ky
pp
E u vE v E vE v v
up
vx v y v v h
Eu Ev E u u E
vp
vy v x v h v
xl
l
σ τ
+−
+−
==
∂∂ −
′ += + =
−∂ − ∂ − −
∂∂ −
′ += +
+∂ + ∂ + +
=
(3.4)
32
The traction on the edge at 0 x = are
k
xl
p and
k
yl
p . The normal stress is
k
xl
p . The shear
stress is
k
yl
p . The outward direction has a positive sign; the inward direction defines a
negative sign.
3.2.2.1.2 The line 0 ( ) x i = with clamped boundary conditions
;; 1,2
kk k k
uuv vk n. = == " (3.5)
The initial displacements on the edge along 0 x = are
k
u and
k
v .
3.2.2.1.3 The line 0 ( ) x i = with mixed boundary conditions
The following example is a possibility of the mixed boundary conditions.
From the central finite difference scheme, the x direction stress is:

11
222 2
11 1 1 2
k
kk
k
E u vE v E vE v v
u
vx v y v v h
+−
∂∂ −
′ += + =
−∂ − ∂ − −
xl
p. (3.6)
And initial displacement is ;1,2
kk
vvk n = = ". (3.7)
3.2.2.2 Right nodes
3.2.2.2.1 The procedure on () x Liii = is similar with the left nodes
The successive discussion will make form like following,

11 1
,,1 ,1 ,1 ,
11 1
22
1
,1
1
(1 )
k
k
kk k k
kk kk kk kk kk
kk k k
kx
k
kk k
k ky
uu u u
AB B C C
vv v v
uf
u
vh
CP
v vf E
ρ
ρ
−+ −
−+ −
−+ −
+
+
+
′′ ′ ′
⎡ ⎤ ⎡⎤ ⎡ ⎤ ⎡⎤ ⎡
++ + +
⎢ ⎥ ⎢⎥ ⎢ ⎥ ⎢⎥ ⎢
′′ ′ ′
⎣ ⎦ ⎣⎦ ⎣ ⎦ ⎣⎦ ⎣
−+ ⎡⎤
⎡⎤ −
+=− +
⎢⎥
⎢⎥
−+
⎢⎥ ⎣⎦
⎣⎦

k
k
u
v
⎤
⎥
⎦
n
(3.8)
Using forward or backward finite difference scheme only is very difficult to deal with
boundary terms. This current case uses the central finite difference scheme throughout
the whole mesh points, and use forward and backward finite difference scheme on the
term. This technique avoids the extra unknown terms. 1 and k =
33
Forward finite difference scheme for boundary condition of elasticity problems in the
Cartesian coordinate systems,

1
1
22 2 2
;
2(1 ) 2(1 ) 2(1 ) 2(1 )
11 1 1
k
k
xy xb y yb
kk
kxb
kk
ky
pp
Eu Ev E u u E
vp
vy v x v h v
E u vE v E vE v v
up
vx v y v v h
b
τ σ
+
+
==
∂∂ −
′ += + =
+∂ + ∂ + +
∂∂ −
′ += + =
−∂ − ∂ − −
(3.9)
Backward finite difference scheme for boundary condition of elasticity problems in the
Cartesian coordinate systems,

1
1
222 2
2(1 ) 2(1 ) 2(1 ) 2(1 )
11 1 1
k
k
kk
ky
kk
kx
Eu Ev E u u E
vp
vy v x v h v
E u vE v E vE v v
up
vx v y v v h
−
−
∂∂ −
′ += + =
+∂ + ∂ + +
∂∂ −
′ += + =
−∂ − ∂ − −
t
t
(3.10)
3.2.2.3 Bottom mesh lines
On the mesh lines ( 0
th
0 ( ) y ii = and
th
n () y Hiv = ), there are several boundary
conditions. On the first mesh line, it contains the 0
th
and the 1
st
mesh points. On the n
th

mesh line, it contains the n
th
and the n+1
th
mesh points. Different boundary conditions
cause different representations. This dissertation shows three type boundary conditions
along these two special edge mesh lines. The following subsections will show how to
deal with the different type boundary conditions on the edge mesh lines ( 0 ( ) y ii =
and ( ) y Hiv = ). There are two catalogues. One is on the 1
st
mesh line and the other one is
on the n
th
mesh line. Each mesh line has three type boundary conditions.
3.2.2.3.1 The line 0 ( ) y ii = with traction boundary condition
In general, the normal stress and the shear stress act on the edge. Follow the traction
boundary conditions definition and use forward finite difference scheme. Collect all
34
derivate and non-derivate terms on the same side, and then make integration on both
sides. Finally, the two important terms and can get from the fundamental computation.
These two important terms substitute into the 1 Navier’s equations. They can remove
the extra finite difference unknown terms, such as and in Navier’s equations.
0
u
0
v
st
0
u
0
v

0
0
10
0
2
10
0
2(1 )
1
b
b
x
y
uu v
vp
hE
vv v
vu p
hE
− +
′ +=
− −
′+=
(3.11)

0
0
10
0
00
2
10
0
00
2(1 )
()
(1 )
()
b
b
LL
x
LL
y
uu v
vdx p dx
hE
vv v
vudx p dx
hE
− +
′ =− +
−−
′ =− +
∫∫
∫∫
(3.12)

00
22
01 1
2 2 22 22 2 2
2(1 ) (1 )
() () () ()
bb
22
x y
LhL vhL vh
uu v p
Lvh Lvh Lvh E Lvh E
+−
=+ − −
−− − −
L
p (3.13a)

00
22 2
01 1
22 2 2 22 2 2
2(1 ) (1 )
() () () ()
bb
2
x y
vhL L v v h L v hL
vu v p
Lvh Lvh Lvh E Lvh E
+−
=+ − −
−− − −
p (3.13b)
3.2.2.3.2 The line 0 ( ) y ii = with clamped boundary conditions

111
(,0) ; (,0)
1 x y
ux u v x v = = (3.14)
The subscript 1 means the 1
st
mesh line. This 1
st
mesh line is located at , so the 0 y = x
and y direction components of coordinate location are and . The initial
displacements are
1
(,0) ux
1
(,0) vx
1 x
u and
1 y
v .

35

3.2.2.3.3 The line 0 ( ) y ii = with mixed boundary condition
This section has many kinds of possibilities. The current example shows one of these
possibilities. This computation get the two terms and to eliminate the extra unknown
terms in the 1 Navier’s equations.
0
u
0
v
st

0
0
,
xb
uu =
0
0, u ′ = (3.15)

0
10
0
2(1 )
b
x
uu v
v
hE
p
− +
′ += (3.16)

0
10
0
00
2(1 )
(
b
LL
x
uu v
vdx p dx
hE
)
− +
′ =− +
∫∫
(3.17)

0
0
1
0
2(1 )
b
xb
x
uu
v
vL
hE
Lp
−
+
=+ (3.18)
3.2.2.4 Upper mesh lines
3.2.2.4.1 The line () y Hiv = with traction boundary condition
In general, the normal stress and shear stress act on the edge. Follow the traction
boundary conditions definition and use backward finite difference scheme. Collect all
derivate and non-derivate terms on the same side, and then make integration on both
sides. Finally, the two important terms
1 n
u
+
and
1 n
v
+
can get from the basic computation.
These two important terms are going to substitute into the Navier’s equations. They
can remove the extra finite difference unknown terms, such as and in Navier’s
equations.
th
n
1 n
u
+ 1 n
v
+

1
1
1
1
2
1
1
2(1 )
1
n
n
nn
nx
nn
ny
uu v
vp
hE
vv v
vu p
hE
t
t
+
+
+
+
+
+
− +
′ +=
− −
′+=
(3.19)
36

1
1
1
1
00
2
1
1
00
2(1 )
()
1
()
n
n
LL
nn
nx
LL
nn
ny
uu v
vdx p dx
hE
vv v
vu dx p dx
hE
+
+
+
+
+
+
− +
′=+
− −
′=+
∫∫
∫∫
t
t
(3.20)

11
22
1
22 22 22 22
2(1 ) (1 )
() ()
nn
nn n xt
LhL vhL vhL
uu v p
Lvh L vh Lvh E Lvh E
22
yt
p
+ +
+
+−
=− + −
−− − −
(3.21a)

11
22 2
2
1
22 22 2 2 22
2(1 ) (1 )
() ()
nn
nn n xt
vhL L v vh v h
vu v Lp L
Lvh L vh Lvh E Lvh E
yt
p
+ +
+
−+ −
=+ − +
−− − −
(3.21b)
3.2.2.4.2 The line () y Hiv = with clamped boundary condition
(, ) ; (, ) .
nxnn
uxnh u v xnh v
yn
= = (3.22)
The initial displacements are
xn
u and
yn
v , so ( , ) 0; ( , ) 0
nn
u x nh v x nh ′ ′ = = .
The subscript n means the nth mesh line. This mesh line is located on , so the yH = x
and y direction coordinate components show (, )
nxn
uxnh u
37
= and (, )
nyn
vxnh v = . On the n
th

mesh line, the
xn
u and
yn
v are the initial displacement. Those coefficients are constants, so
their derivates are zero.
3.2.2.4.3 The line () y Hiv = with mixed boundary condition
This section has many kinds of possibilities. The current example shows one of these
possibilities. Both the displacement and traction boundary conditions are applied. Finally,
this computation got the two terms
1 n
u
+
and
1 n
v
+
to eliminate the extra unknown terms in
the Navier’s equations.
th
n

1
1
,
n
nxt
uu
+
+
= ;
1
0
n
u
+
′ = (3.23)

1
1
1
2(1 )
n
nn
n
uu v
v
hE
xt
p
+
+
+
− +
′ =− + (3.24)

1
1
1
00
2(1 )
(
n
LL
nn
n
uu v
vdx p dx
hE
+
+
+
)
xt
− +
′=+
∫∫
(3.25)

1
1
1
2(1 )
n
n
nxt
n
uu
v
vL
hE
+
xt
Lp
+
+
−
+
=+ (3.26)

3.3 Proposed Method in the polar coordinate systems
3.3.1 Finite difference discretization in the polar coordinate systems
The current case contains n mesh points; mesh lines (Fig. 2.2). The displacement on
mesh line located at
n
th
k ( , ) kh θ defines . The subdivision height between two mesh
points is . The total difference between inner radius and outer radius is . The
displacement on the direction is . The displacement on the
k
u
h H
r
k
u θ direction is . The Finite
difference method along r direction applies on the Navier’s governing equations in the
polar coordinate systems (2.26).
k
v
(, ) ( , ); ( , ); /
kk
uukr ukh v vkh r h Hk θ θθ =Δ = = Δ= = (3.27)
The Central finite difference scheme for Navier’s equations in the polar coordinate
systems is introduced.

11 11
22 2
11
22222
11 11
22
21 1
()
12(1)22(1)
1(3)1
()
12 2(1) 1
21 1
()
2(1 ) 2(1 ) 2 1
1
(
2(1 )
k
kk k k k
k
kk k
kkr
kk k k k
k
k
Eu u u E v v E
u
vh vr h vr
Eu u vE Eu
vuF
vr h v r v r
Ev v v E u u E
v
vh vr h vr
Ev
vr
ρ
−+ −+
−+
−+ −+
2
0
′ ′ −+ − +
′′ ++
−− +
−+ −
′ +− −−+
−− −
′′ −+ − +
′′ ++
+− −
−
+
+
=
11
22 2
(3 ) 1
)0
2 2(1 ) 2(1 )
k
kk
kk
vvE Ev
uv
hvr vr
θ
ρ
−+
+−
′ F + −−+
−+
=
(3.28)

38
3.3.2 Boundary conditions in the polar coordinate systems
In this dissertation, there are four-side boundary conditions for two-dimensional elastic
problems. The proposed method concerns the OB & the AD sides first in Fig. 3.2. After
that, it regards as the other two opposite sides, the arc OA& the arc BD sides. For
example, the notation TC means that edge is traction ( with FC
θ
( ) i ) T θ direction on the
edge. The outward direction is positive sign. The inward direction is negative. The
edge is clamped . The edge is free . The edge is clamped . The
detail information of the mesh lines and the polar coordinate component is in Figure 2.2.
The dissertation is going to investigate the middle dotted line .
( ) ii ( ) C ( ) iii ( ) F ( ) iv ( ) C
( ) m ""

Figure 3.2 an example of a circular continuum with boundary conditions.
The first part considerations want to make form such as ( ) (0, ) ( ) ( , ) ( ) Mtt Nt t Rt η η +Θ= .

11 1 1
11 1 1
(,) [ (,), (,) ( ,), ( ,),
( , ), ( , ), ( , ), ( , )]; 0 or
nn
nn
mt u mt v mt u mt v mt
u mtv mt u mtv mt m
η
−−
−−
=
′′ ′ ′ = Θ
" " (3.29)

39
3.3.2.1 Right nodes
3.3.2.1.1 The line 0 ( ) iii θ = with traction boundary condition
Using forward or backward finite difference scheme only is very difficult to deal with
boundary terms. This current case uses the central finite difference scheme throughout
the whole mesh points, and use forward and backward finite difference scheme on the
term. This technique avoids the extra unknown terms. 1 and k = n
The central finite difference scheme applies the traction boundary conditions,

11
11
22 2 2
;
11
() ( )
2(1 ) 2(1 ) 2
11
() ( )
11 1 2 1
k
k
rrr r
kk k
kr
kk k
kr
pp
Eu vv E v v v
up
vr r r vr h r
Eu vE u v E u u vE u
vp
vr v r r v h v r r
θθ θ
r
θ
τ σ
θ
θ
+−
+−
==
∂∂ −
′ +− = + − =
+∂ ∂ +
∂∂ −
′ ++ = + + =
−∂ − ∂ − −
(3.30)
The traction on the edge at 0 θ = are
k
rr
p and
k
r
p
θ
. The normal stress is
k
r
p
θ
. The shear
stress is
k
rr
p . The outward direction has a positive sign. The inward direction defines a
negative sign.
3.3.2.1.2 The line 0 ( ) iii θ = with clamped boundary condition
;;1,2
rk rk rk rk
uu v v k .n = == " (3.31)
The initial displacements on the edge along 0 r = are
rk
u and
rk
v .
3.3.2.1.3 The line 0 ( ) iii θ = with mixed boundary conditions
The following example is a possibility of the mixed boundary conditions.
From the central finite difference scheme, the θ direction stress is:

11
22 2 2
11
() ( ) ( )
11 1 2 1
k
kk k
k
vE u E v u vE u u E u
v
vr v r r v h v r r
r
p
θ
θ
+−
∂∂ −
′ ++= + +
−∂ − ∂ − −
= . (3.32)
And initial displacement is ;1,2
k
kr
uu k n = = ". (3.33)
40
3.3.2.2 Left nodes
3.3.2.2.1 The procedure on line /2 ( ) i θ π = is similar with the right nodes
The successive discussion will make form like following,

11 1
,,1 ,1 ,1 ,
11 1
22 2
1
,1
1
(1 )
k
k
kk k k
kk kk kk kk kk
kk k k
kr
k
kk k
k k
uu u u
AB B C C
vv v v
uf
u
vrh
CP
v vf E
θ
ρ
ρ
−+ −
−+ −
−+ −
+
+
+
′′ ′ ′
⎡ ⎤ ⎡⎤ ⎡ ⎤ ⎡⎤ ⎡
++ + +
⎢ ⎥ ⎢⎥ ⎢ ⎥ ⎢⎥ ⎢
′′ ′ ′
⎣ ⎦ ⎣⎦ ⎣ ⎦ ⎣⎦ ⎣
−+ ⎡⎤
⎡⎤ −
+=− +
⎢⎥
⎢⎥
−+
⎢⎥ ⎣⎦
⎣⎦

k
k
u
v
⎤
⎥
⎦
(3.34)
3.3.2.3 Bottom mesh lines
On the mesh lines ( and 0
th
0
() rr ii =
th
n
0
() rr H iv = + ), there are several boundary
conditions. On the first mesh line, it contains the 0
th
and the 1
st
mesh points. On the n
th

mesh line, it contains the n
th
and the n+1
th
mesh points. Different boundary conditions
cause different representations. This dissertation shows three type boundary conditions
along these two special edge mesh lines. The following subsections will show how to
deal with the different type boundary conditions on the edge mesh lines
( and ). There are two catalogues. One is on the 1
st
mesh line and
the other one is on the n
th
mesh line. Each mesh line has three type boundary conditions.
0
() rr ii =
0
() rr H iv =+
3.3.2.3.1 The arc with Traction boundary condition
0
() rr ii =
In general, the normal stress and the shear stress act on the edge.
Follow the traction boundary conditions definition and use forward finite difference
scheme. Collect all derivate and non-derivate terms on the same side, and then make
integration on both sides. Finally, the two important terms and can get from the
fundamental computation. These two important terms substitute into the 1 Navier’s
equations.
0
u
0
v
st
41
They can remove the extra finite difference unknown terms, such as and in
Navier’s equations.
0
u
0
v

0
0
10 0
0 22
10 0
0
1
11
1
2(1 )
rb
b
uu u EvE
vp
vh v r r
vv v E
up
vr h r
θ
− ⎛⎞
′ ++ =
⎜⎟
−−
⎝⎠
− ⎛⎞
′+− =
⎜⎟
+
⎝⎠
(3.35)

0
0
2
0100
0100
(1 )
2(1 )
ff
ii
ff
ii
rb
b
vrh
vh v d ru ru vhu p d
E
vrh
hud rv rv hv p d
E
θθ
θθ
θθ
θ
θθ
θ θ
θ θ
⎛⎞ −
′ =− + − +
⎜⎟
⎝⎠
+ ⎛⎞
′ =− + + +
⎜⎟
⎝⎠
∫∫
∫∫
(3.36)

00
2
01
2222
22 2
22 2
()
[( )( )] [( )( )]
( )(1 ) 2(1 )
[ ( )( ) ] [ ( )( ) ]
rb b
rhr vrh
uu
vh r vh r h vh r vh r h
rh v rh vvrh
1
2
v
p p
vh r vh r h E vh r vh r h E
θ
θθ
θθ
θθ
θθ
+
=− −
−− + −− +
+− +
++
−− + −− +
(3.37a)

00
2
01
2222
22 2
22 2
()
[( )( )] [( )( )]
(1 ) 2( )(1 )
[ ( )( ) ] [ ( )( ) ]
rb b
rh r vh r
vu
vh r vh r h vh r vh r h
vrh r vh vrh
1
2
v
p p
vh r vh r h E vh r vh r h E
θ
θθ
θθ
θθ
θθ
−
=− −
−− + −− +
−−+
++
−− + −− +
(3.37b)
3.3.2.3.2 The arc with clamped boundary condition
0
() rr ii =

1
10 1 0
(, ) ; (, )
r
ur u v r v
1
θ
θ θ = = (3.38)
The subscript 1 means the 1
st
mesh line. This 1
st
mesh line is located at , so the r
and
0
rr =
θ direction components of coordinate location are
10
(, ) ur θ and
10
(, ) vr θ . The initial
displacements are
1
r
u and
1
v
θ
.

42
3.3.2.3.3 The arc with mixed boundary condition
0
() rr ii =
This section has many kinds of possibilities. The current example shows one of these
possibilities. This computation get the two terms and to eliminate the extra unknown
terms in the 1 Navier’s equations.
0
u
0
v
st

0
10 0
0
22
1
11
rb
Eu u vE u
vp
vh v r r
− ⎛⎞
′ ++ =
⎜⎟
−−
⎝⎠
(3.39)

0
0 rb
uu = (3.40)

0
2
0100
(1 ) ff
ii
rb
vrh
vh v d ru ru vhu p d
E
θθ
θθ
θ θ
⎛⎞ −
′ =− + − +
⎜⎟
⎝⎠
∫∫
(3.41)

0
2
01
() (1 )
rb rb
rrvh vr
vu u
vh vh vE
θθ−−
=− + +
0
p
θ
(3.42)
3.3.2.4 Upper mesh lines
3.3.2.4.1 The arc with traction boundary condition
0
() rr H iv =+
In general, the normal stress and shear stress act on the edge. Follow the traction
boundary conditions definition and use backward finite difference scheme. Collect all
derivate and non-derivate terms on the same side, and then make integration on both
sides. Finally, the two important terms
1 n
u
+
and
1 n
v
+
can get from the basic computation.
These two important terms substitute into the Navier’s equations. They can remove
the extra finite difference unknown terms, such as
th
n
1 n
u
+
and
1 n
v
+
in Navier’s equations.

1
1
2
11
11
(1 )
()
2(1 )
()
n
n
nnn
nnn
vrh
vhv r vh u ru p
E
vrh
hu r h v rv p
E
θ
rt
t
+
+
++
++
−
′ =− − + +
+
′ =− + + +
(3.43)
43

1
1
2
11
11
(1 )
()
2(1 )
()
ff
n
ii
ff
n
ii
nnn
nnn
vrh
vh v d r vh u ru p d
E
vrh
hu d r hv rv p d
E
θθ
θθ
θθ
θ
θθ
rt
t
θ θ
θ θ
+
+
++
++
⎛⎞ −
′ =−− + +
⎜⎟
⎝⎠
+ ⎛⎞
′ =−+ + +
⎜⎟
⎝⎠
∫∫
∫∫
(3.44)

11
2
1
22 22
22 2
22 22
()
[( )( )] [( )( )]
( )(1 ) 2(1 )
[( )( )] [( )( )]
nn
nn
rt t
rhr vrh
uu
vh r vh r h vh r vh r h
rh v rh vvrh
n
v
p p
vh r vh r h E vh r vh r h E
θ
θθ
θθ
θθ
θθ
+ +
+
−
=− +
−+ − −+ −
−− +
−+
−+ − −+ −
(3.45a)

11
2
1
22 22
22 2
22 22
()
[( )( )] [( )( )]
(1 ) 2( )(1 )
[( )( )] [( )( )]
nn
nn
rt t
rh r vh r
vu
vh rvh rh vh rvh rh
vrh r vh vrh
n
v
p p
vh r vh r h E vh r vh r h E
θ
θθ
θθ
θθ
θθ
+ +
+
+
=−
−+ − −+ −
−++
+−
−+ − −+ −
(3.45b)
3.3.2.4.2 The arc with clamped boundary condition
0
() rr H iv =+
(, ) ; (, )
nrnn
unh u v nh v
n θ
θ θ = = (3.46)
The initial displacements are
rn
u and
n
v
θ
, so ( , ) 0; ( , ) 0
nn
unh v nh θ θ ′ ′ = = .
The subscript n means the nth mesh line. This mesh line is located on , so the
and
0
rr H =+
r θ direction coordinate components show (, )
nrn
unh u
44
θ = and (, )
nn
vnh v θ =
θ
. On the
n
th
mesh line, the initial displacements are
rn
u and
n
v
θ
. Those coefficients are constants, so
their derivates are zero.
3.3.2.4.3 The arc with mixed boundary condition
0
() rr H iv =+
This section has many kinds of possibilities. The current example shows one of these
possibilities. Both the displacement and traction boundary conditions are applied. Finally,
this computation got the two terms
1 n
u
+
and
1 n
v
+
to eliminate the extra unknown terms in
the Navier’s equations.
th
n

1
1
n
nrt
uu
+
+
= ;
1
0
n
u
+
′ = (3.47)

1
2
11
(1 )
()
n
nnn
vrh
vhv r vh u ru p
E
rt
+
++
−
′ =− − + + (3.48)

1
2
11
(1 )
()
ff
n
ii
nnn
vrh
vh v d r vh u ru p d
E
θθ
θθ
rt
θ θ
+
++
⎛⎞ −
′ =−− + +
⎜⎟
⎝⎠
∫∫
(3.49)

1
2
1
() (1 )
n
nrt n
rvh r v r
vu u
vh vh vE
θθ θ
1n
rt
p
+ +
+
−− −
=++ (3.50)

3.4 Assembly of Governing Equations and Boundary Conditions
The boundary conditions of top and bottom sides plug into the Navier’s governing
equation. If the boundary conditions are fixed or free, the matrix will change a little bit,
such as and .
C
1,1
C
, nn
C
Bottom boundary conditions by forward finite difference scheme substitute into the1
st

mesh line of the Navier’s governing equations. This procedure removes extra unknown
coefficients and . The matrices form shows,
0
u
0
v
(3.51)
1
1
1
12 1 2
1,1 1,2 1,1 1,2 1
12 1 2 1
x
y
uf
u uuu
AB C C
vv v v vf
ρ
ρ
−+ ⎡⎤ ′′ ′
⎡⎤ ⎡ ⎤ ⎡⎤ ⎡ ⎤
++ + =Ξ
⎢⎥
⎢⎥ ⎢ ⎥ ⎢⎥ ⎢ ⎥
′′ ′ −+
⎢⎥ ⎣⎦ ⎣ ⎦ ⎣⎦ ⎣ ⎦
⎣⎦

P+
For the Cartesian coordinate systems, the constant Ξ is ; for the polar
coordinate systems, the constant
22
(1 ) / vh E −−
Ξ is .
22 2
(1 ) / vrh E −−
The matrices for the Cartesian coordinate systems demonstrate in
the Appendix A.1. The matrices for the polar coordinate systems
demonstrate in the Appendix A.6.
1,1 1,2 1,1 1,2 1
, , , , and AB C C P
1,1 1,2 1,1 1,2 1
, , , , and AB C C P
45
The mesh lines of the Navier’s governing equations by central finite
difference scheme present as following,
2 ~ 1
nd th
n −
46
k
k
u
v
⎤
⎥
⎦
(3.52)
11 1
,,1 ,1 ,1 ,
11 1
1
,1
1
k
k
kk k k
kk kk kk kk kk
kk k k
kx
k
kk k
k ky
uu u u
AB B C C
vv v v
uf
u
CP
v vf
ρ
ρ
−+ −
−+ −
−+ −
+
+
+
′′ ′ ′
⎡ ⎤ ⎡⎤ ⎡ ⎤ ⎡⎤ ⎡
++ + +
⎢ ⎥ ⎢⎥ ⎢ ⎥ ⎢⎥ ⎢
′′ ′ ′
⎣ ⎦ ⎣⎦ ⎣ ⎦ ⎣⎦ ⎣
−+ ⎡⎤
⎡⎤
+=Ξ +
⎢⎥
⎢⎥
−+
⎢⎥ ⎣⎦
⎣⎦

The matrices for the Cartesian coordinate
systems show in Appendix A.2. The matrices
, ,1 ,1,1, ,1
, , , , , , and
kk kk kk kk kk k k k
AB B C C C P
−+ − +
,,1 ,1 ,1 , ,1
, , , , , , and
kk kk k k kk k k kk k
A BB C C C
−+ − +
P
P+

for the polar coordinate systems show in Appendix A.7.
Top boundary conditions by backward finite difference scheme substitute into the
mesh line of the Navier’s governing equations. This procedure deletes extra unknown
coefficients and . The matrices form becomes,
th
n
1 n
u
+ 1 n
v
+
(3.53)
11
,,1 ,1 ,
11
n
n
nx
nn n n
nn nn nn nn n
nn n n ny
uf
uu u u
AB C C
vv v v vf
ρ
ρ
−−
−−
−−
−+ ⎡⎤ ′′ ′
⎡ ⎤ ⎡⎤ ⎡⎤ ⎡ ⎤
++ + =Ξ
⎢⎥
⎢ ⎥ ⎢⎥ ⎢⎥ ⎢ ⎥
′′ ′ −+
⎢⎥ ⎣ ⎦ ⎣⎦ ⎣⎦ ⎣ ⎦
⎣⎦

The matrices for the Cartesian coordinate systems
demonstrate in Appendix A.3. The matrices for the polar
coordinate systems demonstrate in Appendix A.8.
,,1 ,1 ,
, , , ,and
nn nn n n nn n
AB C C P
−−
,,1 ,1 ,
, , , ,and
nn nn n n nn n
AB C C P
−−

3.5 State Space Formulations

2( ) 2( ) 2( ) 1
;, , ; ,
nn n
AU BU CU U P A B C U P
× × ″′
++ =Ξ+ ∈ ∈

\ \ (3.54)
What the new simple form governing equation is combine boundary conditions with
Navier’s governing equations by finite difference scheme along y or axis. r
Where the state variables vectors is defined,
[ ]
11 2 2
(,) ( ,) (,) ( ,) (,) ( ,)
T
nn
U u gt v gt u gt v gt u gt v gt = " (3.55)
For the Cartesian coordinate systems, the letter is g x .
For the polar coordinate systems, the letter is g θ .
After rearranged the state variables vectors
[ ]
T
UU η ′ = with Laplace transformation, the
equation (3.54) becomes the first order ordinary differential equation.

4( ) 4( ) 4( ) 1
(, ) ( ) ( , ) (, ); ; ,
nn n
d
gs F s gs q g s F q
dg
ηη η
× ×
=+ ∈ ∈ \\ (3.56)

12 1
0
;
()
I
F
AC I A B ω
− −⎥ 1
0
. q
AP
−
⎡⎤
=
⎢
−+Ξ −
⎣⎦
⎡ ⎤
=
⎢ ⎥
⎣ ⎦
(3.57)

Since the top and bottom boundary conditions substitute into the Navier’s governing
equations, those procedures construct a new first order ordinary differential equation.
The two-dimensional elasticity problems have four-side boundary conditions. Now it is
about to deal with the other two boundary conditions, left-side and right-side.

4( ) 1
() (0, ) () ( , ) ( ); ,
n
Ms sNsms Rs R ηη η
×
+= ∈ \ (3.58)

The matrices M and for the Cartesian coordinate systems are in Appendix A.4 and A.5;
the matrices
N
M and for the polar coordinate systems represent in Appendix A.9 and
A.10.
N
47
(3.59a)
12 4( ) 4( )
2( ) 2( )
(1,2)
() ; ( ) ;
00
,0
sub sub nn
nn
subi i
MM
Ms Ms
M
×
×
=
⎡⎤
=∈
⎢⎥
⎣⎦
∈
\ \ (3.59b)
4( ) 4( )
34
2( ) 2( )
(3.4)
00
() ; () ;
,0
nn
sub sub
nn
subi i
Ns Ns
NN
N
×
×
=
⎡⎤
=∈
⎢⎥
⎣⎦
∈
\ \
3.6 Solution by DTFM
After the strain-displacement relations and Hooke’s laws substitute into the equilibrium
equations, the Navier’s governing equations grained. The finite difference method
separates the Navier’s governing equations along y or direction, then top and bottom
boundary conditions plug into the semi-finite difference Navier’s governing equations.
Those procedures make A, B, C, and P matrices. The other two side boundary conditions
build M, N and R matrices. By rearranging the state variables those matrices become F, q
with the same M, N, and R matrices. Applying distributed transfer function method on
those two equations, the closed form result shows in (2.39). If the system is static, the
time parameter is equal to zero.
r

3.7 Assembly of Multiple Rectangular Subsections
The proposed method concerns the region I ABCDE first in Fig. 3.3. After that, it regards
as the region II CDGF. For example, the notation means that edge is
clamped traction( . The edge is free . The edge is free . The edge( is
traction with
xx
CFFT FT F ( ) i
) C () ii () F () iii () F ) iv
() T x direction on the edge. The outward direction is positive sign. The
48
inward direction is negative. The edge is free( . The edge( is traction with ( ) v ) F ) vi ( ) T x
direction on the edge. The edge( is free . The detail information of the mesh lines
and the Cartesian coordinate component is in Figure 2.3. The dissertation is going to
investigate the middle dotted line ( .
) vii ( ) F
) m ""

Figure 3.3 an example of an L-shaped continuum with boundary conditions.

Ij Ij Ij Ij Ij Ij
KKHR σ η = = ,
(3.60)
2( ) 8() 8() 8() 2( ) 1 8()1
;; ;,
nn n n n n
Ij Ij Ij Ij Ij
KR H R R ση
×× ×
∈∈ ∈ ∈ \ ×
\ R
IIj IIj IIj IIj IIj IIj
KKH σ η = = ,
(3.61)
2( )4( ) 4( )4( ) 2( ) 1 4( ) 1
;; ;,
nn nn n n
IIj IIj IIj IIj IIj
KH R ση
×× ×
∈∈ ∈ ∈ \\ \ ×
\ [ ]
T
cb
R RR = ; [ ]
cb
HH H = (3.62)
The relationship between stress σ and displacement η is stiffness matrix denotes .
The term comes from the finite difference distributed transfer function method. The
subscript
i
K
HR
or I II indicates the two different sub-dominated regions. The subscript j
49
means the middle mesh points of whole structure in the Figure 2.3. The subscript c
means the interconnected mesh points. The subscript b means the known boundary
conditions. It is easy to find out interconnected mesh points displacement from the force
balance relationship along the interconnected mesh point.

Ij Ijc Ijc Ij Ijb Ijb IIj IIjc IIjc IIj IIjb IIjb
KH R KH R K H R K H R + =+ (3.63)

3.8 Static Numerical Examples
Several numerical examples of finite difference distributed transfer function method on
two-dimensional elastic continua present next. The example 3.1 is 2-D elasticity analysis
on square region with boundary conditions in the Cartesian coordinates systems.
The example 3.2 is 2-D elasticity analysis on square region with boundary
conditions in the Cartesian coordinates systems. The example 3.3 is 2-D elasticity
analysis on annular sector with T boundary conditions in the polar coordinate
systems. The example 3.4 is 2-D elasticity analysis on annular sector with TC
boundary conditions in the polar coordinate systems. The example 3.5 is 2-D elasticity
analysis on L-shaped region with boundary conditions. The example 3.6 is
2-D elasticity analysis on one-shaped region with CFC boundary conditions. The
choice of material is steel. The density of steel is 7850 . The modulus of elasticity
is 200GP . The Possion’s ratio is 0.32 (Hibberler, 1994). The length and height are1 .
The in-plane external force is1 N . The longest height in L-shaped continua is .
y
CCCT
x
TCFC
CCC
θ
FC
θ
xx
CFFT FT F
TxFF
3
/ Kg m
a m
k 2 m

50
Example 3.1

Figure 3.4 a square region with boundary conditions
y
CCCT

This example shows the investigation of FDDTFM method on a in the Cartesian
coordinates systems. The notation region means that edge is clamped ,
edge is clamped , edge
y
CCCT
y
CCCT 0 x = () C
0 y = () C x L = is clamped( , and edge is subjected to a
vertical uniform load of magnitude
) C yH =
p () . The investigation of the on A-B line
shows that FDDTFM 160 mesh lines have very close approach with FEM 729 elements.
The Figure 3.5~3.9 shows the 6, 20, 80 and 160 mesh lines convergent tendency. The
FDDTFM convergence is guaranteed.
Ty
y
CCCT
51

0 0.1 0.2 0.3 0.4 0.5 0.6 0.7 0.8 0.9 1
-5
-4
-3
-2
-1
0
1
2
3
4
5
x 10
-14
δ/P
x
C C C T
y
(2D elasticity)

δ
x
FDDTFM 6
δ
x
FDDTFM 20
δ
x
FEETFM 80
δ
x
FDDTFM 160
δ
x
FEM 27X27

Figure 3.5 distribution of displacement
x
δ on A-B for the square region
y
CCCT

0 0.1 0.2 0.3 0.4 0.5 0.6 0.7 0.8 0.9 1
-0.5
0
0.5
1
1.5
2
2.5
3
3.5
4
x 10
-13
δ/P
x
C C C T
y
(2D elasticity)

δ
y
FDDTFM 6
δ
y
FDDTFM 20
δ
y
FDDTFM 80
δ
y
FDDTFM 160
δ
x
FEM 27X27

Figure 3.6 distribution of displacement
y
δ on A-B for the square region
y
CCCT
52

0 0.1 0.2 0.3 0.4 0.5 0.6 0.7 0.8 0.9 1
0.02
0.03
0.04
0.05
0.06
0.07
0.08
0.09
0.1
x
σ/P
C C C T
y
(2D elasticity)

σ
x
FDDTFM 6
σ
x
FDDTFM 20
σ
x
FDDTFM 80
σ
x
FDDTFM 160
σ
x
FEM 27X27

Figure 3.7 distribution of stress
x
σ on A-B for the square region
y
CCCT

0 0.1 0.2 0.3 0.4 0.5 0.6 0.7 0.8 0.9 1
0.02
0.04
0.06
0.08
0.1
0.12
0.14
0.16
0.18
0.2
0.22
x
σ/P
C C C T
y
(2D elasticity)

σ
y
FDDTFM 6
σ
y
FDDTFM 20
σ
y
FDDTFM 80
σ
y
FDDTFM 160
σ
y
FEM 27X27

Figure 3.8 distribution of stress
y
σ on A-B for the square region
y
CCCT
53

0 0.1 0.2 0.3 0.4 0.5 0.6 0.7 0.8 0.9 1
-0.2
-0.15
-0.1
-0.05
0
0.05
0.1
0.15
x
τ/P
C C C T
y
(2D elasticity)

τ
xy
FDDTFM 6
τ
xy
FDDTFM 20
τ
xy
FDDTFM 80
τ
xy
FDDTFM 160
τ
xy
FEM 27X27

Figure 3.9 distribution of shear stress
xy
τ on A-B for the square region
y
CCCT

Table 3.1 displacements and stresses of the square region at (
y
CCCT 0.5,0.5)

Method Mesh
13
10
y
δ ×
x
σ
y
σ
6 3.222 0.024 0.1947
20 3.712 0.02177 0.2069
80 3.934 0.02148 0.2134
FDDTFM
160 3.976 0.02153 0.2147
FEM 27 27 × 3.989 0.02135 0.2148
Difference
160
729
1 100
FDDTFM
FEM
−×
0.32% 0.84% 0.05%

At middle point ( on A-B line denotes( , the 0.5, 0.5) xy == ) C x axis displacement and
shear stress
xy
τ are almost the same point. Table 3.1 shows the difference between
54
FDDTFM (160) and FEM (729) is very small. The y axis displacement error is 0.32%,
the stress
x
σ error is 0.84%, and the stress
y
σ error is 0.05%.

Example 3.2
This example shows the investigation of FDDTFM method on a in the Cartesian
coordinates systems. The notation region means that edge is subjected to a
horizontal uniform load of magnitude
x
TCFC
x
TCFC 0 x =
p () , edge Tx 0 y = is clamped , edge ( ) C x L = is
free end( , and edge is clamped( . The investigation of the on A-B
line shows that FDDTFM 155 mesh lines have very close approach with FEM 900
elements. The Figure 3.11~3.15 shows the 5, 15, 29 and 155 mesh lines convergent
tendency. The FDDTFM convergence is guaranteed.
) F yH = ) C
x
TCFC

Figure 3.10 a square region with boundary conditions
x
TCFC
55

0 0.1 0.2 0.3 0.4 0.5 0.6 0.7 0.8 0.9 1
1
1.5
2
2.5
3
3.5
4
4.5
x 10
-12
δ/P
x
T
x
C F C (2D Elasticity-Cartesian)

δ
u
FDDTFM 5
δ
u
FDDTFM 15
δ
u
FDDTFM 29
δ
u
FDDTFM 155
δ
u
FEM 30x30

Figure 3.11 distribution of displacement
x
δ on A-B for the square region
x
TCFC

0 0.1 0.2 0.3 0.4 0.5 0.6 0.7 0.8 0.9 1
-5
-4
-3
-2
-1
0
1
2
3
4
5
x 10
-12
δ/P
x
T
x
C F C (2D Elasticity-Cartesian)

δ
v
FDDTFM 5
δ
v
FDDTFM 15
δ
v
FDDTFM 29
δ
v
FDDTFM 155
δ
v
FEM 30x30

Figure 3.12 distribution of displacement
y
δ on A-B for the square region
x
TCFC
56

0 0.1 0.2 0.3 0.4 0.5 0.6 0.7 0.8 0.9 1
-1.2
-1
-0.8
-0.6
-0.4
-0.2
0
0.2
x
σ/P
T
x
C F C (2D Elasticity-Cartesian)

σ
x
FDDTFM 5
σ
x
FDDTFM 15
σ
x
FDDTFM 29
σ
x
FDDTFM 155
σ
x
FEM 30x30

Figure 3.13 distribution of stress
x
σ on A-B for the square region
x
TCFC

0 0.1 0.2 0.3 0.4 0.5 0.6 0.7 0.8 0.9 1
-1
-0.8
-0.6
-0.4
-0.2
0
0.2
0.4
x
σ/P
T
x
C F C (2D Elasticity-Cartesian)

σ
y
FDDTFM 5
σ
y
FDDTFM 15
σ
y
FDDTFM 29
σ
y
FDDTFM 155
σ
y
FEM 30x30

Figure 3.14 distribution of stress
y
σ on A-B for the square region
x
TCFC
57

0 0.1 0.2 0.3 0.4 0.5 0.6 0.7 0.8 0.9 1
-0.5
-0.4
-0.3
-0.2
-0.1
0
0.1
0.2
0.3
0.4
0.5
x
σ/P
T
x
C F C (2D Elasticity-Cartesian)

τ
xy
FDDTFM 5
τ
xy
FDDTFM 15
τ
xy
FDDTFM 29
τ
xy
FDDTFM 155
τ
xy
FEM 30x30

Figure 3.15 distribution of stress
xy
τ on A-B for the square region
x
TCFC

Table 3.2 displacements and stresses of the square region at (0.5, 0.5)
x
TCFC

Method Mesh
12
10
u
δ ×
x
σ
y
σ
5 2.322 -0.449 -0.065
15 1.982 -0.460 -0.054
29 1.891 -0.460 -0.052
FDDTFM
155 1.795 -0.458 -0.051
FEM 30 30 × 1.764 -0.456 -0.050
Difference
155
900
1 100
FDDTFM
FEM
−×
1.8% 0.4% 2%

At middle point ( on A-B line denotes , the 0.5, 0.5) xy == ( ) C y axis displacement and
shear stress
xy
τ are almost the same point. Table 3.2 shows the difference between
FDDTFM (155) and FEM (900) is very small. The x axis displacement error is 1.8%,
the stress
x
σ error is 0.4%, and the stress
y
σ error is 2%.
58
Example 3.3
This example shows the investigation of FDDTFM method on a T in the Polar
coordinates systems. The notation TC region means that edge
CCC
θ
CC
θ
2
π
θ = is subjected to
a horizontal uniform load of magnitude p () T
θ
, edge
i
rr = is clamped( , edge ) C 0 θ = is
clamped , and edge is clamped . The investigation of the TC on A-B
line shows that FDDTFM 15 mesh lines have very close approach with FEM 1500
elements. The Figure 3.17~3.21 shows the 6, 16, and 42 mesh lines convergent tendency.
() C
o
rr = () C CC
θ

Figure 3.16 an annular region with TC boundary conditions CC
θ
59
0 pi/6 pi/3 pi/2
-8
-6
-4
-2
0
2
4
6
8
x 10
-12
δ/P
θ
T
θ
C C C (2D Elasticity-Polar)

δ
r
FDDTFM 5
δ
r
FDDTFM 10
δ
r
FDDTFM 15
δ
r
FEM 30X50

Figure 3.17 distribution of displacement
r
δ on A-B for the TC annular region CC
θ
0 pi/6 pi/3 pi/2
0
0.5
1
1.5
2
2.5
3
3.5
4
x 10
-12
δ/P
θ
T
θ
C C C (2D Elasticity-Polar)

δ
θ
FDDTFM 6
δ
θ
FDDTFM 16
δ
θ
FDDTFM 42
δ
θ
FEM 30X50

Figure 3.18 distribution of displacement
θ
δ on A-B for the T annular region CCC
θ

60

0 pi/6 pi/3 pi/2
-1
-0.8
-0.6
-0.4
-0.2
0
0.2
θ
σ/P
T
θ
C C C (2D Elasticity-Polar)

σ
r
FDDTFM 6
σ
r
FDDTFM 16
σ
r
FDDTFM 38
σ
r
FEM 30X50

Figure 3.19 distribution of stress
r
σ on A-B for the TC annular region CC
θ
0 pi/6 pi/3 pi/2
-1.2
-1
-0.8
-0.6
-0.4
-0.2
0
θ
σ/P
T
θ
C C C (2D Elasticity-Polar)

σ
θ
FDDTFM 6
σ
θ
FDDTFM 10
σ
θ
FDDTFM 16
σ
θ
FEM 30X50

Figure 3.20 distribution of stress
θ
σ on A-B for the TC annular region CC
θ

61
0 pi/6 pi/3 pi/2
-8
-6
-4
-2
0
2
4
6
8
θ
τ/P
T
θ
C C C (2D Elasticity-Polar)

τ
r θ
FDDTFM 5
τ
r θ
FDDTFM 10
τ
r θ
FDDTFM 15
τ
r θ
FEM 30X50

Figure 3.21 distribution of stress
r θ
τ on A-B for the TC annular region CC
θ

Table 3.3 displacements and stresses of the T annular region at (1.5 CCC
θ
, / 3 π )

Method Mesh
12
10
θ
δ ×
θ
σ
r
σ
6 1.041 -0.352 -0.0189
16 0.862 -0.321 -0.0257
FDDTFM
42 0.793 -0.0296/38
FEM 30 50 × 0.757 -0.320 -0.0299
Difference
42
1500
1 100
FDDTFM
FEM
−×
4.75% 0.3% 1%

At specific point (1.5,
3
r )
π
θ == on A-B line denotes( , the ) C θ axis displacement and
shear stress
r θ
τ are almost the same point. Table 3.3 shows the difference between
FDDTFM (42) and FEM (1500) is very small. The θ axis displacement error is 4.75%,
the stress
θ
σ error is 0.3%, and the stress
r
σ error is 1% based on 38 mesh lines.
62
Example 3.4

Figure 3.22 an annular region with TC boundary conditions FC
θ

This example shows the investigation of FDDTFM method on a TC in the Polar
coordinate system. The notation TC region means that edge
FC
θ
FC
θ
2
π
θ = is subjected to a
horizontal uniform load of magnitude p () T
θ
, edge
i
rr = is clamped , edge ( ) C 0 θ = is
free , and edge is clamped . The investigation of the TC on A-B line
shows that FDDTFM 87 mesh lines have very close approach with FEM 1500 elements.
The Figure 3.23~3.27 shows the 5, 15, 29 and 87 mesh lines convergent tendency.
( ) F
o
rr = ( ) C FC
θ
63
0 pi/6 pi/3 pi/2
-1
0
1
2
3
4
5
6
7
8
9
x 10
-11
δ/P
θ
T
θ
C F C (2D Elasticity-Polar)

δ
r
FDDTFM 5
δ
r
FDDTFM 15
δ
r
FDDTFM 29
δ
r
FDDTFM 87
δ
r
FEM 30X50

Figure 3.23 distribution of displacement
r
δ on A-B for the TC annular region FC
θ
0 pi/6 pi/3 pi/2
0
0.5
1
1.5
2
2.5
3
3.5
4
x 10
-12
δ/P
θ
T
θ
C F C (2D Elasticity-Polar)

δ
θ
FDDTFM 5
δ
θ
FDDTFM 15
δ
θ
FDDTFM 29
δ
θ
FDDTFM 87
δ
θ
FEM 30X50

Figure 3.24 distribution of displacement
θ
δ on A-B for the TC annular region FC
θ
64
0 pi/6 pi/3 pi/2
-0.8
-0.7
-0.6
-0.5
-0.4
-0.3
-0.2
-0.1
0
0.1
θ
σ/P
T
θ
C F C (2D Elasticity-Polar)

σ
r
FDDTFM 8
σ
r
FDDTFM 12
σ
r
FDDTFM 22
σ
r
FDDTFM 32
σ
r
FEM 30X50

Figure 3.25 distribution of stress
r
σ on A-B for the TC annular region FC
θ
0 pi/6 pi/3 pi/2
-1.2
-1
-0.8
-0.6
-0.4
-0.2
0
0.2
θ
σ/P
T
θ
C F C (2D Elasticity-Polar)

σ
θ
FDDTFM 5
σ
θ
FDDTFM 15
σ
θ
FDDTFM 29
σ
θ
FDDTFM 87
σ
θ
FEM 30X50

Figure 3.26 distribution of stress
θ
σ on A-B for the TC annular region FC
θ
65
0 pi/6 pi/3 pi/2
-10
-8
-6
-4
-2
0
2
4
6
8
10
θ
τ/P
T
θ
C F C (2D Elasticity-Polar)

τ
r θ
FDDTFM 5
τ
r θ
FDDTFM 15
τ
r θ
FDDTFM 29
τ
r θ
FDDTFM 87
τ
r θ
FEM 30X50

Figure 3.27 distribution of stress
r θ
τ on A-B for the annular region TCFC
θ

Table 3.4 displacements and stresses of the TC annular region at (1 FC
θ
.5, / 3 π )
Method Mesh
12
10
θ
δ ×
θ
σ
r
σ
5 1.150 -0.375 -0.022/6
15 0.881 -0.324 -0.024/12
29 0.820 -0.027/22
FDDTFM
87 0.775 -0.029/32
FEM 30 50 × 0.758 -0.319 -0.029
Difference
87
1500
1 100
FDDTFM
FEM
−×
2.2% 1.6% 0%

At specific point (1.5,
3
r )
π
θ == on A-B line denotes( , the ) C θ axis displacement and
shear stress
r θ
τ are almost the same point. Table 3.4 shows the difference between
66
FDDTFM (87) and FEM (1500) is very small. The θ axis displacement error is 2.2%,
the stress
θ
σ error is 1.6%, and the stress
r
σ error is 0% based on 32 mesh lines.

Example 3.5

Figure 3.28 an L-shaped region with boundary conditions
x
CFFT FFF
This example shows the investigation of FDDTFM method on a in the
Cartesian coordinates systems. The notation region means that edge AH is
clamped . The edge is free . The edge GF is free . The edge is
subjected to a horizontal uniform load of magnitude
x
CFFT FFF
x
CFFT FFF
( ) C HG ( ) F ( ) F FE
P( )
x
T . The edge is free ( .
The edge DB is clamped . The edge
ED ) F
( ) C BA is free . The investigation of the
on G-D line shows that FDDTFM much fewer mesh lines have very close
approach with FEM 8112 elements. The Figure 3.29~3.32 shows the results due to
several mesh lines convergent tendency.
( ) F
x
CFFT FFF

67
Table 3.5 displacements and stresses of the square region at (1, 0.5)
x
CFFT FFF
Method
12
10
u
δ ×
12
10
v
δ ×
x
σ
xy
τ
5.696/8+4 3.142/14+7 1.036/10+5 0.050/10+5
5.095/10+5 2.677/20+10 1.014/12+6 0.040/12+6
FDDTFM
4.825/14+7 2.329/40+20
FEM
10452 5252 ×+ ×
4.800 2.322 1.020 0.042
Difference 0.5% 0.3% 0.6% 5%

0 0.1 0.2 0.3 0.4 0.5 0.6 0.7 0.8 0.9 1
1
2
3
4
5
6
7
8
9
10
x 10
-12
δ/P
y
C F F T
x
F F F (2D elasticity L-shaped)

δ
u
FDDTFM 8+4
δ
u
FDDTFM 10+5
δ
u
FDDTFM 14+7
δ
u
FEM 104X52+52X52

Figure 3.29 distribution of displacement
u
δ on G-D for the L-shaped region
x
CFFT FFF
68
0 0.1 0.2 0.3 0.4 0.5 0.6 0.7 0.8 0.9 1
1.5
2
2.5
3
3.5
4
x 10
-12
δ/P
y
C F F T
x
F F F (2D elasticity L-shaped)

δ
v
FDDTFM 14+7
δ
v
FDDTFM 20+10
δ
v
FDDTFM 40+20
δ
v
FEM 104X52+52X52

Figure 3.30 distribution of displacement
v
δ on G-D for the L-shaped region
x
CFFT FFF
0 0.1 0.2 0.3 0.4 0.5 0.6 0.7 0.8 0.9 1
0.8
1
1.2
1.4
1.6
1.8
2
σ/P
C F F T
x
F F F(2D elasticity L-Shaped)

σ
x
FDDTFM 10+5
σ
x
FDDTFM 12+6
σ
x
FDDTFM 90+45
σ
x
FEM 104X52+52X52

Figure 3.31 distribution of stress
x
σ on G-D for the L-shaped region
x
CFFT FFF
69
0 0.1 0.2 0.3 0.4 0.5 0.6 0.7 0.8 0.9 1
-0.5
-0.4
-0.3
-0.2
-0.1
0
0.1
0.2
σ/P
C F F T
x
F F F(2D elasticity L-Shaped)

τ
xy
FDDTFM 10+5
τ
xy
FDDTFM 12+6
τ
xy
FDDTFM 90+45
τ
xy
FEM 104X52+52X52

Figure 3.32 distribution of stress
xy
τ on G-D for the L-shaped region
x
CFFT FFF

At specific point ( on G-D line denotes . Table 3.5 shows the difference
between FDDTFM (less than 90 mesh lines) and FEM (8112) is very small. The
difference error % defines
1, 0.5) xy == ( ) C
90
8112
1 100
FDDTFM
FEM
<
−× . The x axis displacement error is
0.5%, the y axis displacement error is 0.3%, the stress
x
σ error is 0.6%, and the stress
xy
τ error is 5%.

The non-dimensional displacements display as
2
1
u
u
E
vP
δ
δ =
−
and
2
1
v
v
E
vP
δ
δ =
−
. The non-
dimensional stresses represent as
x
x
P
σ
σ = ,
y
y
P
σ
σ = and
xy
x
P
τ
τ = .
70
Example 3.6

Figure 3.33 a square region with boundary conditions
x
CFCT FF

The notation region (Figure 3.33) means that edge
x
CFCT FF 0 x = is clamped , edge
&
() C
0 y = 0
2
L
x ≤≤ is free , edge ( ) F 0 y = &
2
L
x L ≤ ≤ is clamped , edge ( ) C x L = is
subjected to a horizontal uniform load of magnitude p() Tx , edge & yH =
2
L
x L ≤≤ is
free , and edge ( ) F y H = & 0
2
L
x ≤ ≤ is free . The investigation of the on
A-B line shows that results of FDDTFM 180 mesh lines are very close to the results of
FEM 10,000 elements. The following figures (Figure 3.34~3.37) show the 20, 60, 100
and 180 mesh lines convergent tendency. The FDDTFM convergence is guaranteed.
( ) F
x
CFCT FF
71

0 0.1 0.2 0.3 0.4 0.5 0.6 0.7 0.8 0.9 1
-0.1
0
0.1
0.2
0.3
0.4
0.5
0.6
0.7
δ
y
C F C T
x
F F(2D elasticity)

δ
u
FDDTFM 60
δ
u
FDDTFM 80
δ
u
FDDTFM 140
δ
u
FEM 100X100

Figure 3.34 distribution of displacement
u
δ on A-B of the square region
x
CFCT FF

0 0.1 0.2 0.3 0.4 0.5 0.6 0.7 0.8 0.9 1
-0.25
-0.2
-0.15
-0.1
-0.05
0
0.05
0.1
δ
y
C F C T
x
F F(2D elasticity)

δ
v
FDDTFM 60
δ
v
FDDTFM 80
δ
v
FDDTFM 140
δ
v
FEM 100X100

Figure 3.35 distribution of displacement
v
δ on A-B of the square region
x
CFCT FF

72

0 0.1 0.2 0.3 0.4 0.5 0.6 0.7 0.8 0.9 1
-2
-1.5
-1
-0.5
0
0.5
1
1.5
σ/P
C F C T
x
F F(2D elasticity)
y

σ
x
FDDTFM 60
σ
x
FDDTFM 80
σ
x
FDDTFM 140
σ
x
FEM 100X100

Figure 3.36 distribution of stress
x
σ on A-B of the square region
x
CFCT FF

0 0.1 0.2 0.3 0.4 0.5 0.6 0.7 0.8 0.9 1
-0.1
-0.05
0
0.05
0.1
0.15
0.2
0.25
x
σ
y
/P
C F C T
x
F F (2D elasticity) on D-E

σ
y
FDDTFM 60
σ
y
FDDTFM 80
σ
y
FEM 100X100

Figure 3.37 distribution of shear stress
y
σ on D-E of the square region
x
CFCT FF

73
Table 3.6 displacements and Stresses of the square region at
x
CFCT FF C
AB − DE − ; δ σ (mesh lines)
Method
u
δ
v
δ
x
σ
y
σ
0.3835(60) -0.0134(60) 0.7803(60) -0.0414(60)
0.3888(80) -0.0158(80) 0.7912(80) -0.0428(80)
FDDTFM
0.3926(140)-0.0182(140)0.7979(140)
FEM(100 ) 100 × 0.3994 -0.0191 0.8092 -0.0431
Difference
140
10000
1100
FDDTFM
FEM
−×

1.7% 4.9% 1.4% 0.7%

At middle point ( on A-B line denotes , the Table 3.6 shows the
difference between FDDTFM and FEM is very small. The
0.5, 0.5) xy == ( ) C
x axis displacement
u
δ error is
1.7%, the y axis displacement
v
δ error is 4.9%, the stress
x
σ error is 1.4%, and the stress
y
σ error on D-E line at middle point (C) is 0.7%.

3.9 Dynamic Numerical examples
Several numerical approaches of finite difference distributed transfer function method on
the plate show next. The example 3.7 is 2-D elasticity analysis on rectangular region with
boundary condition in the Cartesian coordinate systems. The height is 1.2 and the
width is 1. The example 3.8 is 2-D elasticity analysis on annular sector region with
boundary conditions in the polar coordinate systems. The outer radius is 2 and
the inner radius is 1. The following results show the non-dimensional number.
CCCC
CCCC

74
Example 3.7
This example shows the investigation of FDDTFM method on a CC in the Cartesian
coordinates system. The notation CCCC region means that edge is clamped ,
edge is clamped ( , edge
CC
0 x = ( ) C
0 y = ) C x L = is clamped , and edge ( ) C y H = is clamped ( . ) C
The investigation of the CCCC on natural frequency of free vibration shows that
FDDTFM 100 mesh lines have very close approach with FEM 10000 elements.
Table 3.7 natural frequency of aCCCC 2-D elastic rectangular region
FDDTFM FEM % Δ Mode no.
8 50 100 30 30 × 100 100 ×
1 0.52290.56100.56570.566540.56596 0.05%
2 0.55390.56660.56820.566790.56601 0.10%
3 0.64280.67250.67490.676730.67455 0.05%
4 0.78260.82210.82630.829320.82608 0.03%
5 0.85030.92790.93360.937700.93343 0.02%
6 0.87150.94050.93620.944650.93936 0.12%
7 0.93220.94100.94260.948020.94010 0.10%
8 1.02251.06201.06901.0761 1.0692 0.05%
9 1.05211.12551.13431.1431 1.1342 0.01%
10 1.06061.13111.13491.1438 1.1343 0.05%

The Table 3.7 shows the 8, 50, and 100 mesh lines convergent tendency. The FDDTFM
convergence is guaranteed. Those results also compare with FEM. The definition of
is the closest difference between FDDTFM and FEM on the Table. The result shows
the average is less than 0.058%.
% Δ
% Δ

Example 3.8

This example shows the investigation of FDDTFM method on a CCCC in the polar
coordinates system. The notation CC region means that edge is clamped , CC
0
90 θ = () C
75
edge is clamped , edge
i
rr = () C
0
0 θ = is clamped , and edge is clamped .
The investigation of the CCCC on natural frequency of free vibration shows that
FDDTFM 16 mesh lines have very close approach with FEM 1800 elements.
() C
o
rr = () C
Table 3.8 natural frequency of aCCCC 2-D elastic annular sector region

FDDTFM FEM % Δ Mode no.
4 10 16 816 × 30 60 ×
1 0.32610.35600.36520.362970.36082 1.21%
2 0.42490.46890.48040.483780.48018 0.05%
3 0.46420.51270.52650.527240.52099 1.06%
4 0.48840.52420.53800.528070.52355 0.12%
5 0.53280.58180.59330.582520.57433 0.12%
6 0.55090.60020.61640.627520.61560 0.13%
7 0.61950.65250.66710.672610.66138 0.86%
8 0.65780.65410.71310.686730.67364 2.99%
9 0.67090.70850.72000.732280.71328 0.67%
10 0.69100.77300.78690.800750.78482 0.27%

The Table 3.8 shows the 4, 10, and 16 mesh lines convergent tendency. The FDDTFM
convergence is guaranteed. Those results also compare with FEM. The definition of
is the closest difference between FDDTFM and FEM on the Table. The result shows
the average is less than 0.748%.
% Δ
% Δ

3.10 Discussion
In the example 3.1, it is easy to check the boundary condition with zero displacement in
Figure 3.5 and Figure 3.6. The absolute value of difference between FDDTFM and FEM
divided by FEM defines as an error (difference). In the example 3.2, it is straightforward
to check boundary condition on the left edge with
2
()
1
x
Eu v
vp
vx y
σ
∂∂
= +=−
−∂ ∂
and
76
boundary on the right edge with free edge
2
()
1
x
Eu v
v
vx y
σ 0
∂ ∂
= +=
−∂ ∂
. (i.e., Figure 3.13).
In the example 3.3, the boundary conditions on the clamped edge with zero displacement
are in Figure 3.17 and Figure 3.18. The boundary condition on the edge,
0
90
0
90 θ
σ
=
is
2
1
(
1
Eu vu
v
vrr r θ
∂∂
++ =−
−∂ ∂
)p (i.e., Figure 3.20). In the example 3.4, the boundary
conditions on the edge,
0
90
0
90 θ
σ
=
is
2
1
(
1
Eu vu
v
vrr r θ
)p
∂ ∂
+ +=−
−∂ ∂
and the boundary
conditions on the edge,
0
0
0
0 θ
σ
=
is
2
1
(
1
Eu vu
v
vrr r θ
)0
∂ ∂
+ +=
−∂ ∂
. This situation displays in
Figure 3.26. From now on, the investigation on the boundary condition can provide
confirmative solution case by case. In the example 3.5, the stress concentration factor
shows on the corner apparently. (i.e., Figure 3.31 and 3.32) In the example 3.6, even
there are different boundary condition along the same side, the FDDTFM (140 mesh lines)
shows the faster convergence than FEM (10,000 elements) (i.e., Figure 3.34 ~ 3.37). The
computation of natural frequency by FDDTFM is fewer mesh lines than by FEM mesh
elements. (Table 3.7~3.8)

77
Chapter 4

Steady and Unsteady State Heat Conduction Problems in Two-
dimensional Region

4.1 Problem Statement
This dissertation presents an innovative method called finite difference distributed
transfer function method (FDDTFM) to solve the heat conduction problems in two-
dimensional region. The heat conduction equation (2.7) derived from the conservation of
energy (2.5; 2.6). The proposed method uses three major boundary conditions (2.17; 2.18;
2.19) to solve two-dimensional rectangular, circular, and L-shaped heat conduction
problems.

4.2 Proposed Method in the Cartesian coordinate Systems
4.2.1 Finite difference discretization in the Cartesian coordinate systems
The current case contains n mesh points; mesh lines (Fig. 2.1). The temperature on
mesh line located at
n
th
k ( , ) x kh defines . The subdivision height between two mesh
points is . The total height of the two-dimensional heat conduction region is . The
total width of the two-dimensional heat conduction region is .
k
T
h H
L
(, ) ( , );
k
H
T T xk y T xkh y h
k
=Δ= Δ== (4.1)
78
The temperature on the x direction is . The Finite difference method along
k
T y direction
applies on the two-dimensional heat conduction governing equations in the Cartesian
coordinate systems (2.7).
The Central finite difference scheme for two-dimensional heat conduction governing
equations in the Cartesian coordinate systems is introduced.

11
2
2
kk k
k
TTT
kT g cT
h
ρ
+−
−+ ⎛⎞
′′
++
⎜⎟
⎝⎠
= (4.2)
4.2.2 Boundary conditions in the Cartesian coordinate systems
In this dissertation, there are four-side boundary conditions for two-dimensional heat
conduction problems.

Figure 4.1 an example of rectangular heat conduction with boundary conditions
The proposed method concerns the OB & the ED sides first in Fig. 4.1. After that, it
regards as the other two opposite sides, the OE & the BD sides. The notation
means that edge THAC 0 ( ) x i = is prescribed temperature on the edge. The edge ( ) T
79
0 ( ) y ii = is prescribed heat flux . The heat supply into the body direction is positive
sign. The heat supply removal from the body direction is negative. The edge
( ) H
( ) x L iii =
is adiabatic . The edge ( ) A ( ) y Hiv = is convection . The detail information of the
mesh lines and the Cartesian coordinate component is in Figure 2.1. The dissertation is
going to investigate the middle dotted line .
( ) C
( ) m ""
The first part considerations want to make form such as ( ) (0, ) ( ) ( , ) ( ) Mtt Nt Lt Rt η η += .

12 1
12 1
(,) [ (,), (,) ( ,), ( ,),
( , ), ( , ) ( , ), ( , )]; 0 or
nn
nn
mt T mtTmt T mtT mt
T mtTmt T mtT mt m L
η
−
−
=
′′ ′ ′ =
" " (4.3)
4.2.2.1 Left nodes
4.2.2.1.1 The line 0 ( ) x i = with prescribed temperature boundary condition

00
(, , )
x l
Tx yt T
=
= (4.4)
The prescribed temperature boundary condition on the edge at 0 x = is . For the finite
mesh points, .
0l
T
0
(0)
kl
TT =
4.2.2.1.2 The line 0 ( ) x i = with prescribed heat flux boundary condition

0
(, , )
0 x flux l
Tx yt
k q
x
=
∂
− =
∂
(4.5)
The conduction heat flow into the body at 0 x = equals to external prescribed heat
flux . If there is no flux across the boundary surface (i.e., ), it is so
called adiabatic boundary conditions. The prescribed heat flux is
0
at 0
flux l
qx =
0
0
flux l
q =
0 flux l
q . For the finite
mesh points, .
0
(0) /
kflux
Tq ′ =−
l
k

80
4.2.2.1.3 The line 0 ( ) x i = with convection boundary condition
The conduction heat flow into the body at 0 x = equals to convection heat flux from the
fluid at temperature T to the surface at prescribed temperature . The heat
transfer coefficient is . For the finite mesh points,
0
at 0
l
Tx =
c
h
0
(0) (0)/ /
kck cl
ThT khTk ′ − += .

00
(, , )
(
xc l
Tx yt
kh )TT
x
=
∂
− =−
∂
(4.6)
4.2.2.2 Right nodes
4.2.2.2.1 The procedure on line () x Liii = is similar with the left nodes
The successive discussion will make form as following,
[] [ ] [] [ ]
2
,,11 ,,11 kkkkk k kkkkk k k
ch
AT C T C T C T T P
k
ρ
−− + +
′′ ⎡⎤ ++ + −
⎣⎦
k
=
n
i
(4.7)
Using forward or backward finite difference scheme only is very difficult to deal with
boundary terms. This current case uses the central finite difference scheme throughout
the whole mesh points, and use forward and backward finite difference scheme on the
term. This technique avoids the extra unknown terms. On the mesh lines
( and ), there are several boundary conditions. On the first
mesh line, it contains the 0
th
and the 1
st
mesh points. On the n
th
mesh line, it contains the
n
th
and the n+1
th
mesh points. Different boundary conditions cause different
representations. This dissertation shows three type boundary conditions along these two
special edge mesh lines. The following subsections will show how to deal with the
different type boundary conditions on the edge nodal lines (
1 and k =
0
th
0 ( ) yi =
th
n () yH iv =
0 ( ) y ii = and ( ) y Hiv = ).
There are two catalogues. One is on the 1
st
mesh line and the other one is on the n
th
mesh
line. Each mesh line has three type boundary conditions.
81
4.2.2.3 Bottom nodal lines
Follow the heat flux or convection boundary conditions definition and use forward finite
difference scheme. Finally, an important term gets from the fundamental computation.
This important term substitutes into the 1
st
two-dimensional heat conduction equation.
This procedure can remove the extra finite difference unknown term, such as in 1
st

two-dimensional heat conduction equation.
0
T
0
T
4.2.2.3.1 The line 0 ( ) y ii = with prescribed temperature boundary condition

00
(, , )
y
Tx yt T
=
=
b
(4.8)
The prescribed temperature boundary condition on the edge at 0 y = is . For the finite
mesh points, .
0b
T
00b
TT =
4.2.2.3.2 The line 0 ( ) y ii = prescribed heat flux boundary condition

1
0
(, , )
kk
y
Tx yt T T
kk
yh
+
=
∂−
−=−
∂
0fluxb
q= (4.9)
The conduction heat flow into the body at 0 y = equals to external prescribed heat
flux . If there is no flux across the boundary surface (i.e., ), it is so
called adiabatic boundary conditions. The prescribed heat flux is
0
at 0
flux b
qy =
0
0
flux b
q =
0 flux b
q . For the finite
mesh point, .
01 0
/
flux b
TT q hk =+
4.2.2.3.3 The line with convection boundary condition 0 ( ) yi = i

1
00 0
(, , )
() ; (
kk
yc b c b
Tx yt T T
k hTT k hTT
yh
+
=
∂−
−=−− =
∂
)
k
− (4.10)
82
The conduction heat flow into the body at 0 y = equals to convection heat flux from the
fluid at temperatureT to the surface at prescribed temperature . For the finite
mesh points, . The heat transfer coefficient is .
0
at 0
b
Ty =
10 0 0
() /
cb c
T T hhT k hhT k −− = − /
c
h

01
c
b
cc
khh
TT
khh k hh
=+
++
0
T (4.11)
4.2.2.4 Upper nodal lines
Follow the heat flux or convection boundary conditions definition and use backward
finite difference scheme. Finally, an important term
1 n
T
+
gets from the fundamental
computation. This important term substitutes into the n
th
two-dimensional heat
conduction equation. This procedure can remove the extra finite difference unknown term,
such as in n
th
two-dimensional heat conduction equation.
1 n
T
+
4.2.2.4.1 The line () y Hiv = with prescribed temperature boundary condition

0
(, , )
yH t
Tx yt T
=
= (4.12)
The prescribed temperature boundary condition on the edge at y H = is . For finite
mesh points,
0t
T
0 n
TT
t
=
4.2.2.4.2 The line with prescribed heat flux boundary condition () yH iv =

1
0
(, , )
kk
y H flux t
Tx yt T T
kk
yh
−
=
∂−
==
∂
q (4.13)
The conduction heat flow into the body at y H = equals to external prescribed heat
flux . If there is no flux across the boundary surface (i.e., ), it is so
0
at
flux t
qy =H
0
0
flux t
q =
83
called adiabatic boundary conditions. The prescribed heat flux is
0 flux t
q . For finite mesh
points, .
10
/
nn fluxt
TT q h
+
=+ k
4.2.2.4.3 The line with convection boundary condition () yH iv =
The conduction heat flow into the body at yH = equals to convection heat flux from the
fluid at temperature T to the surface at prescribed temperature . The heat
transfer coefficient is . For finite mesh points,
0
at
t
Ty H =
c
h
11
//
nn cn c t
TT hhT k hhT
++ 0
k − =− + .

1
0
(, , )
() ; (
kk
yH ct ct k
Tx yt T T
khTTk h
yh
−
=
∂−
=− =−
∂
0
)TT (4.14)

1
c
nn
cc
khh
TT
khh k hh
+
=+
+−
0t
T (4.15)
4.3 Proposed Method in the polar coordinate Systems
4.3.1 Finite difference discretization in the polar coordinate systems
The current case contains n mesh points; mesh lines (Fig. 2.2). The temperature on
mesh line located at
n
th
k (, ) kh θ defines . The subdivision height between two mesh
points is . The total difference between inner radius and outer radius is .
k
T
h H
(, ) ( , );
k
H
TTkr Tkh r h
k
θθ =Δ = Δ= = (4.16)
The temperature at ( , ) kh θ is . The Finite difference method along r direction applies
on the heat conduction governing equations in the polar coordinate systems (2.28).
k
T
The Central finite difference scheme for two-dimensional heat conduction governing
equations in the polar coordinate systems is introduced.

11 11
22
21 1
2
kk k k k k
k
TTT T T g c
T
hr h r k
k
T
k
ρ
−+ −+
−+ − +
′′ +++=
(4.17)
84
4.3.2 Boundary conditions in the polar coordinate systems
In this dissertation, there are four-side boundary conditions for two-dimensional heat
conduction problems.
The proposed method concerns the OB & the ED sides first in Fig. 4.2. After that, it
regards as the other two opposite sides, the arc OE& the arc BD sides. For example, the
notation means that edge is prescribed temperature ( on the edge. The
edge is prescribed heat flux . The heat supply into the body direction is positive
sign. The heat supply removal from the body direction is negative. The edge is
convection . The edge is adiabatic . The detail information of the mesh lines
and the Cartesian coordinate component is in Figure 2.2. The dissertation is going to
investigate the middle dotted line ( .
THAC ( ) i ) T
( ) ii ( ) H
( ) iii
( ) C ( ) iv ( ) A
) m ""

Figure 4.2 an example of circular heat conduction with boundary conditions

85
The first part considerations want to make form such as ( ) (0, ) ( ) ( , ) ( ) Mtt Nt t Rt η η +Θ= .

12 1
12 1
(,) [ (,), (,) ( ,), ( ,),
( , ), ( , ), ( , ), ( , )]; 0 or
nn
nn
mt T m t T mt T m t T mt
Tmt T mt T mt T mt m
η
−
−
=
′′ ′ ′ = Θ
" " (4.18)
4.3.2.1 Right nodes
4.3.2.1.1 The line 0 ( ) iii θ = with prescribed temperature boundary condition

0
(,0, ) ; 1,2 .
k
rk r
Tr t T k n = = " (4.19)
The prescribed temperature boundary condition on the edge at 0 θ = is .
0r
T
4.3.2.1.2 The line 0 ( ) iii θ = with prescribed heat flux boundary condition

0
(, , )
;1,2
k
rk flux r
Tr t
kkTqk .n
θ
θ
∂
′ −=−= =
∂
" (4.20)
The conduction heat flow into the body at 0 θ = equals to external prescribed heat
flux
0
at 0
flux r
q θ = . The prescribed heat flux is
0 flux r
q .
If there is no flux across the boundary surface (i.e.,
0
0
r
q = ), it is so called adiabatic
boundary conditions.
4.3.2.1.3 The line 0 ( ) iii θ = with convection boundary condition

(, , ) Tr t
k
θ
θ
∂
−
∂

0
()
c
hT T = − ;
0
(0) (0)
c
kk
hh
TT
kk
′ −+ =
c
r
T (4.21)
The conduction heat flow into the body at 0 x = equals to convection heat flux from the
fluid at temperature T to the surface at prescribed temperature . The heat
transfer coefficient is . For the finite mesh points,
0
at 0
l
Tx =
c
h
0
(0) (0) / /
kck cr
ThT khTk ′ − += .

86

4.3.2.2 Left nodes
4.3.2.2.1 The procedure on line /2 ( ) i θ π = is similar with the left nodes
The successive discussion will make form like following,
[] [ ] [] [ ]
22
,,11 ,,11
2
kk k kk k kk k kk k k k
cr h
AT C T C T C T T P
k
ρ
−− + +
′′ ⎡⎤ ++ + −
⎣⎦
=
n
(4.22)
Using forward or backward finite difference scheme only is very difficult to deal with
boundary terms. This current case uses the central finite difference scheme throughout
the whole mesh points, and use forward and backward finite difference scheme on the
term. This technique avoids the extra unknown terms. On the mesh lines
( and ), there are several boundary conditions. On the first
mesh line, it contains the 0
th
and the 1
st
mesh points. On the n
th
mesh line, it contains the
n
th
and the n+1
th
mesh points. Different boundary conditions cause different
representations. This dissertation shows three type boundary conditions along these two
special edge mesh lines. The following subsections will show how to deal with the
different type boundary conditions on the edge nodal lines (i.e.,
1 and k =
0
th
0
() rr ii =
th
n
0
() rr H iv =+
0
() rr ii =
and ). There are two catalogues. One is on the 1
st
mesh line and the other
one is on the n
th
mesh line. Each mesh line has three type boundary conditions.
0
() rr H iv =+
4.3.2.3 Bottom nodal lines
Follow the heat flux or convection boundary conditions definition and use forward finite
difference scheme. Finally, an important term gets from the fundamental computation.
This important term substitutes into the 1
st
two-dimensional heat conduction equation.
0
T
87
This procedure can remove the extra finite difference unknown term, such as in 1
st

two-dimensional heat conduction equation.
0
T
4.3.2.3.1 The arc with prescribed temperature boundary condition
0
() rr ii =

0
0
(, , )
rr b
Tr t T θ
=
= (4.23)
The prescribed temperature boundary condition on the edge at
0
rr = is . For the finite
mesh points, .
0b
T
00b
TT =
4.3.2.3.2 The arc with prescribed heat flux boundary condition
0
() rr ii =
The conduction heat flow into the body at
0
rr = equals to external prescribed heat
flux . If there is no flux across the boundary surface (i.e., ), it is so
called adiabatic boundary conditions. The prescribed heat flux is
0
at
flux b
qr =
0
r
0
0
flux b
q =
0 flux b
q . For the finite
mesh point, .
01 0
/
flux b
TT q hk =+

0
1
0
(, , )
kk
r r flux b
Tr t T T
kk
rh
q
θ
+
=
∂−
−=− =
∂
(4.24)
4.3.2.3.3 The arc with convection boundary condition
0
() rr ii =

0
1
0
(, , )
() ; (
kk
rr cb cb k
Tr t T T
k hTT k hTT
rh
0
)
θ
+
=
∂−
−=−− =
∂
− (4.25)
The conduction heat flow into the body at
0
rr = equals to convection heat flux from the
fluid at temperatureT to the surface at prescribed temperature . For the finite
mesh points, . The heat transfer coefficient is .
0
at
b
Tr =
0
r
/
10 0 0
() /
cb c
T T hhT k hhT k −− = −
c
h

01
c
b
cc
khh
TT
khh k hh
=+
++
0
T (4.26)

88
4.3.2.4 Upper nodal lines
Follow the heat flux or convection boundary conditions definition and use backward
finite difference scheme. Finally, an important term
1 n
T
+
gets from the fundamental
computation. This important term substitutes into the n
th
two-dimensional heat
conduction equation. This procedure can remove the extra finite difference unknown term,
such as in n
th
two-dimensional heat conduction equation.
1 n
T
+
4.3.2.4.1 The arc with prescribed temperature boundary condition
0
() rr H iv =+

0
0
(, , )
rr H t
Tr t T θ
=+
= (4.27)
The prescribed temperature boundary condition on the edge at
0
rr H = + is . For finite
mesh points,
0t
T
0 n
TT
t
=
4.3.2.4.2 The arc with prescribed heat flux boundary condition
0
() rr H iv =+

0
1
0
(, , )
kk
r r H flux t
Tr t T T
kk
rh
q
θ
−
=+
∂−
==
∂
(4.28)
The prescribed heat flux is
0 flux t
q . For finite mesh points,
10
/
nn fluxt
TT q h
+
k = + . If there is
no flux across the boundary surface (i.e.,
0
0
flux t
q = ), it is so called adiabatic boundary
conditions. The conduction heat flow into the body at
0
rr H = + equals to external
prescribed heat flux
00
at
flux t
qrrH = + .
4.3.2.4.3 The arc with convection boundary condition
0
() rr H iv =+

1
0
(, , )
() ; (
kk
ct ct k
Tx H t T T
k hTT k hTT
rh
−
∂−
=− =−
∂
0
) (4.29)
89
The conduction heat flow into the body at y H = equals to convection heat flux from the
fluid at temperature T to the surface at prescribed temperature . The heat
transfer coefficient is . For finite mesh points,
0
at
t
Ty H =
c
h
10
//
nn c t cn
TT hhT khhT
++1
k −=− .

1
c
nn
cc
khh
TT
khh khh
+
=+
++
0t
T (4.30)

4.4 Assembly of Governing Equations and Boundary Conditions
The boundary conditions of top and bottom sides plug into the two-dimensional heat
conduction governing equation. If the boundary conditions are different, the C matrix
will change a little bit, such as and . Bottom boundary conditions by forward
finite difference scheme substitute into the 1
1,1
C
, nn
C
st
mesh line of the two-dimensional heat
conduction governing equations. This procedure removes extra unknown coefficients .
The matrices form shows,
0
T
[ ] [ ] [ ]
1,1 1 1,1 1 1,2 2 1 1
AT C T C T T P ′′ ⎡⎤ + ++Ξ
⎣⎦
= (4.31)
The matrices
1,1 1,1 1,2 1
, , , and A CC P for the Cartesian coordinate systems demonstrate in the
Appendix B.1; the matrices for the polar coordinate systems
demonstrate in the Appendix B.6. For the Cartesian coordinate systems, the constant
is . For the polar coordinate systems, the constant
1,1 1,1 1,2 1
, , , and AC C P
Ξ
2
/ ch k ρ − Ξ is .
22
2/ cr h k ρ −
The mesh lines of the two-dimensional heat conduction governing equations
by central finite difference scheme present as following,
2 ~ 1
nd th
n −
[ ] [ ] [ ] [ ]
,,11 ,,11 kkkkk k kkkkk k k k
A TC T C TC T T P
−− + +
′′ ⎡⎤ ++ + +Ξ
⎣⎦
= (4.32)
90
The matrices for the Cartesian coordinate systems show in
Appendix B.2; the matrices
,,1 , ,1
, , , , and
k k kk kk k k k
AC C C P
−+
,,1 , ,1
, , , , and
k k kk kk k k k
A CC C
−+
P for the polar coordinate systems
show in Appendix B.7.
Top boundary conditions by backward finite difference scheme substitute into the
mesh line of the two-dimensional heat conduction governing equations. This procedure
deletes extra unknown coefficients
th
n
1 n
T
+
. The matrices form becomes,
[ ] [ ] [ ]
,,11 , nn n nn n nn n n n
AT C T C T T P
−−
′′ ⎡⎤ + ++Ξ
⎣⎦
= (4.33)
The matrices for the Cartesian coordinate systems demonstrate in
Appendix B.3; the matrices
,,1 ,
, , ,and
nn n n nn n
AC C P
−
,,1 ,
, , ,and
nn nn nn n
ACC
−
P for the polar coordinate systems
demonstrate in Appendix B.8.

4.5 State Space Formulations
What the new simple form governing equation as following is combine boundary
conditions with two-dimensional heat conduction governing equations by finite
difference scheme along or axis. y r

1
;, ; ,
nn n
AU CU U P A C U P
× × ″
++Ξ = ∈ ∈
\\ (4.34)
Where the state variables vectors is defined,
[ ]
1 234 1
(,) ( ,) (,) ( ,) (,) ( ,)
T
nn
U T gt T gt T gt T gt T gt T g t
−
= " (4.35)
For the Cartesian coordinate systems, the letter is g x . For the polar coordinate systems,
the letter is g θ .
91
After rearranged the state variables vectors [ ]
T
UU η ′ = with Laplace transformation, the
equation (4.34) becomes the first order ordinary differential equation.

2( ) 2( ) 2( ) 1
(, ) ( ) ( , ) (, ); ; ,
nn n
d
gs F s gs q g s F q
dg
ηη η
× ×
=+ ∈ ∈ \\ (4.36)

1
0
;
()0
I
92
F
AC I μ
−
⎡ ⎤
=
⎢ ⎥
−−Ξ
⎦
1
0
. q
AP
−
⎡ ⎤
=
⎢ ⎥
⎣ ⎦
(4.37)
⎣
The solution of equation (4.34) is exponential decreased, so the negative value μ in
equation (4.37) is reasonable. Since the top and bottom boundary conditions substitute
into the two-dimensional heat conduction governing equations, those procedures
construct a new first order ordinary differential equation. The two-dimensional heat
conduction problems have four-side boundary conditions. Now it is about to deal with the
other two boundary conditions, left-side and right-side.

2( ) 1
() (0, ) () ( , ) ( ); ,
n
Ms sNsms Rs R ηη η
×
+= ∈ \ (4.38)
(4.39a)
12 2( ) 2( )
(1,2)
() ; ;
00
,0 ;
sub sub nn
nn
subi i
MM
Ms M
M
×
×
=
⎡⎤
=∈
⎢⎥
⎣⎦
∈
\ \ (4.39b)
2( ) 2( )
34
(3,4)
00
() ; ;
,0 ;
nn
sub sub
nn
subi i
Ns N
NN
N
×
×
=
⎡⎤
=∈
⎢⎥
⎣⎦
∈
\ \ The matrices M and for the Cartesian coordinate systems represent in Appendix B.4
and B.5; the matrices
N
M and for the polar coordinate systems represent in Appendix
B.9 and B.10.
N

4.6 Solution by DTFM
The two-dimensional heat conduction governing equations grained from conservation of
energy. The finite difference method separates the two-dimensional heat conduction
governing equations along y or direction, then top and bottom boundary conditions
plug into the semi-finite difference two-dimensional heat conduction governing equations.
Those procedures make A, B, C, and P matrices. The other two side boundary conditions
build M, N and R matrices. By rearranging the state variables those matrices become F, q
with the same M, N, and R matrices. Applying distributed transfer function method on
those two equations, the closed form result shows in (2.39). If the system is steady state,
the time parameter is equal to zero.
r

4.7 Assembly of Multiple Rectangular Subsections
The proposed method concerns the region I ABCDE first in Fig. 4.3. After that, it regards
as the region II CDGF. For example, the notation TAAHTHC means that edge is
prescribed temperature . The edge is adiabatic( . The edge( is adiabatic( .
The edge is prescribed heat flux . The heat supply into the body direction is
positive sign. The heat supply removal from the body direction is negative. The inward
direction is negative. The edge is prescribed temperature . The edge is
prescribed heat flux . The edge( is convection . The detail information of the
mesh lines and the Cartesian coordinate component is in Figure 2.3. The dissertation is
going to investigate the middle dotted line .
( ) i
( ) T ( ) ii ) A ) iii ) A
( ) iv ( ) H
( ) v ( ) T ( ) vi
( ) H ) vii ( ) C
( ) m ""

Ij Ij Ij Ij
Gq H R η = + ,
93

2( ) 2( ) 2( ) 1
,;,
nn n
Ij Ij Ij Ij
Gq H R η
× ×
∈∈ \ \ (4.40)

IIj IIj IIj IIj
Gq H R η = + ,

2( ) 2( ) 2( ) 1
,;,
nn n
IIj IIj IIj IIj
Gq H R η
× ×
∈\ ∈\ (4.41)
Where
0
() ( , ) ( )
L
ii i
Gq x G x q d ξξ ξ =
∫

[ ]
T
cb
R RR = (4.42)
The terms come from the finite difference distributed transfer function method.
The subscript
& Gq HR
or I II indicates the two different sub-dominated regions. The subscript j
means the middle mesh points of whole structure in the Figure 2.3. The subscript c
means the interconnected mesh points. The subscript b means the known boundary
conditions. It is easy to find out interconnected mesh points from temperature and heat
flux equilibrium.

Ij Ij Ij IIj IIj IIj
Gq H R Gq H R + =+ (4.43)

Figure 4.3 an example of L-shaped heat conduction with boundary conditions
94

4.8 Steady State Numerical Examples
Several numerical examples of finite difference distributed transfer function method on
two-dimensional heat conduction problem present next. The example 4.1 is square region
heat conduction withTTTT boundary conditions in the Cartesian coordinate systems. The
prescribed temperature T along left edge assumes . The prescribed temperature
along bottom edge is . The prescribed temperature T along right edge is .
The prescribed temperature T along top edge assumes . The example 4.2 is square
region heat conduction with THAC boundary conditions in the Cartesian coordinate
systems. The example 4.3 is annular sector heat conduction with TTTT boundary
conditions in the polar coordinate systems. The example 4.4 is annular sector heat
conduction with TH boundary conditions in the polar coordinate systems. The
prescribed temperature T along left edge assumes . The heat
flux
0
20 C
T
0
300 C
0
200 C
0
100 C
AC
0
20 C
flux
q sets . The gas free convection heat transfer coefficient
=10 . The ambient temperature is 100 . The example 4.5 contains L-
shaped heat conduction withTAAHTHC boundary conditions. The choice of material is
carbon steel 1.0%. The thermal conduction =43 . The density of the
body
2
100 / Wm
c
h /( )
o
Wm C ⋅
0
C
k /( )
o
Wm C ⋅
ρ =7801 . The specific heat of material c =0.473 . (Holman, 1986)
3
/ kg m
0
/( ) kJ kg C ⋅

Example 4.1
This example shows the investigation of FDDTFM method on a TTTT in the Cartesian
coordinates systems. The notation TTTT region means that edge is prescribed 0 x =
95
temperature , edge is prescribed temperature , edge ( ) Tp 0 y = ( ) Tp x L = is prescribed
temperature ( , and edge ) Tp y H = is prescribed temperature . ( ) Tp

Figure 4.4 the rectangular heat conduction with TTTT boundary conditions
The investigation of the TTTT on A-B line shows that FDDTFM 61 mesh lines have
very close approach with FEM 100 elements. The Figure 4.5 shows the 5, 15, 25 and 61
mesh lines convergent tendency.

Table 4.1 temperature of the TTTT square region at ( 0.5,0.5)

Method Mesh T
5 144.5
15 151.0
25 152.6
FDDTFM
61 154.0
FEM 10 10 × 155.7
Difference
60
100
1 100
FDDTFM
FEM
−×
0.1%
96

0 0.1 0.2 0.3 0.4 0.5 0.6 0.7 0.8 0.9 1
0
50
100
150
200
250
Temperature
x
T T T T (Heat Conductin-Cartesian)

FDDTFM 5
FDDTFM 15
FDDTFM 25
FDDTFM 61
FEM 10X10

Figure 4.5 distribution of temperature on A-B for the TTTT square region

At middle point ( on A-B line denotes ( . Table 4.1 shows the difference
between FDDTFM (61) and FEM (100) is very small. The error of temperature is 0.1%.
0.5, 0.5) xy == ) C

Example 4.2
This example shows the investigation of FDDTFM method on a TH in the Cartesian
coordinates systems. The notation TH region means that edge is prescribed
temperature ( , edge is heat flux( , edge
AC
AC 0 x =
) Tp 0 y = ) Hf x L = is adiabatic ( , and edge ) Ab
y H = is convection . ( ) Cv

97

Figure 4.6 the rectangular heat conduction with THAC boundary conditions

0 0.1 0.2 0.3 0.4 0.5 0.6 0.7 0.8 0.9 1
19
20
21
22
23
24
25
26
27
28
29
Temperature
x
T H A C (Heat Conductin-Cartesian)

FDDTFM 3
FDDTFM 5
FDDTFM 15
FEM 10X10

Figure 4.7 distribution of temperature on A-B for the TH square region AC

98
The investigation of the TH on A-B line shows that FDDTFM 15 mesh lines have
very close approach with FEM 100 elements. The Figure 4.7 shows the 3, 5, and 15 mesh
lines convergent tendency.
AC

Table 4.2 temperature of the TH square region at AC (0.5,0.5)

Method Mesh T
3 26.01
5 26.04
FDDTFM
15 26.09
FEM 10 10 × 26.1
Difference
15
100
1 100
FDDTFM
FEM
−×
0.003%

At middle point ( on A-B line denotes ( . Table 4.2 shows the difference
between FDDTFM (15) and FEM (100) is very small. The error of temperature is 0.1%.
0.5, 0.5) xy == ) C

Example 4.3
This example shows the investigation of FDDTFM method on a TTTT in the polar
coordinates systems. The notation TTTT region means that edge is prescribed
temperature , edge is prescribed temperature , edge
0 x =
( ) Tp 0 y = ( ) Tp x L = is prescribed
temperature ( , and edge ) Tp y H = is prescribed temperature . The investigation of
the TTTT on A-B line shows that FDDTFM 19 mesh lines have very close approach with
FEM 200 elements. The Figure 4.9 shows the 5, 9, and 19 mesh lines convergent
tendency. The FDDTFM convergence is guaranteed.
( ) Tp
99

Figure 4.8 the annular heat conduction with TTTT boundary conditions

0 pi/6 pi/3 pi/2
20
40
60
80
100
120
140
160
180
200
220
Temperature
θ
T T T T (Heat Conductin-Polar)

FDDTFM 5
FDDTFM 9
FDDTFM 19
FEM 10X20

Figure 4.9 distribution of temperature on A-B for the TTTT annular region

100
Table 4.3 temperature of the TTTT annular region at (1.5, / 3) π

Method Mesh T
5 155.06
9 160.00
FDDTFM
19 163.25
FEM 10 20 × 163.08
Difference
19
200
1100
FDDTFM
FEM
−×
0.01%

At specific point ( 1.5, /3) xy π == on A-B line denotes . Table 4.3 shows the
difference between FDDTFM (19) and FEM (200) is very small. The error of temperature
is 0.01%.
( ) C

Example 4.4

Figure 4.10 the annular heat conduction with TH boundary conditions AC
101

0 pi/6 pi/3 pi/2
20
25
30
35
40
45
50
55
60
Temperature
θ
T H A C (Heat Conductin-Polar)

FDDTFM 5
FDDTFM 19
FDDTFM 35
FEM 10X20

Figure 4.11 distribution of temperature on A-B for the THAC annular region
This example shows the investigation of FDDTFM method on a TH in the polar
coordinates systems. The notation TH region means that edge is prescribed
temperature ( , edge is heat flux( , edge
AC
AC 0 x =
) Tp 0 y = ) Hf x L = is adiabatic ( , and edge ) Ab
y H = is convection . The investigation of the TH on A-B line shows that
FDDTFM 35 mesh lines have very close approach with FEM 200 elements. The Figure
4.11 shows the 3, 5, and 15 mesh lines convergent tendency. The FDDTFM convergence
is guaranteed.
( ) Cv AC
At specific point ( 1.5, /3) xy π == on A-B line denotes . Table 4.4 shows the
difference between FDDTFM (35) and FEM (200) is very small. The error of temperature
is 0.08%.
( ) C

102
Table 4.4 temperature of the THAC annular region at (1.5, / 3) π

Method Mesh T
5 42.28
19 42.48
FDDTFM
35 42.52
FEM 10 20 × 42.89
Difference
35
200
1 100
FDDTFM
FEM
−×
0.08%

Example 4.5

Figure 4.12 the L-shaped heat conduction with T boundary conditions AHHTHC
This example shows the investigation of FDDTFM method on a TAHHT in the
Cartesian coordinates systems. The notation T region means that edge
HC
AHHTHC
AH along is prescribed temperature . The edge is adiabatic () Tp HG ( ) Ab . The edge
is heat flux . The edge is heat flux . The edge GF (Hf) ) FE (Hf ED is prescribed
temperature( . The edge ) Tp DB is heat flux . The edge (Hf) BA is convection . () Cv
103

0 0.1 0.2 0.3 0.4 0.5 0.6 0.7 0.8 0.9 1
20
40
60
80
100
120
140
160
180
Temperature
y
T A H H T H C (Heat Conduction L-shaped)

FDDTFM 100+50
FDDTFM 300+150
FDDTFM 700+350
FEM 120X60+60X60

Figure 4.13 distribution of temperature on A-B for the TAHHT L-shaped region HC

Table 4.5 temperature of the T L-shaped region at AHHTHC (1,1)
Method Mesh T
100+50 68.16
300+150 68.96
FDDTFM
700+350 69.27
FEM 120 60 60 60 × +× 70.13
Difference
1050
10800
1 100
FDDTFM
FEM
−×
1.24%

The investigation of the T on G-D line shows that FDDTFM much fewer mesh
lines have very close approach with FEM 10800 elements. The Figure 4.13 shows the
results due to 1050 mesh lines convergent tendency.
AHHTHC
104
At specific point ( on G-D line denotes . Table 4.5 shows the difference
between FDDTFM (1050) and FEM (10800) is very small. The error of temperature is
1.24%.
1, 0.5) xy == ( ) C

4.9 Unsteady State Numerical examples
Several numerical examples of finite difference distributed transfer function method on
two-dimensional heat conduction problem present next. The example 4.6 is 1-D heat
conduction with boundary conditions in the Cartesian coordinate systems. The
prescribed temperature T along left edge assumes . The heat flux F along right edge
is . The example 4.7 is 1-D heat conduction with TH boundary conditions in
the Cartesian coordinate systems. The example 4.8 is 2-D heat conduction with TTF
boundary conditions in the Cartesian coordinate systems. The example 4.9 is 2-D
composite heat conduction with TTTF boundary conditions in the Cartesian
coordinate systems. The prescribed temperature T along left edge assumes . The heat
flux
TF
0
0 C
2
100 / Wm
T
TT
0
0 C
flux
q on the right edge sets . The choice of material is carbon steel 1.0%.
The thermal conduction k =43
2
100 / Wm
/( )
o
Wm C ⋅ . The density of the body ρ =7801 . The
specific heat of material =0.473 . (Holman, 1986)
3
/ kg m
c
0
/( ) kJ kg C ⋅
Before getting involved in 2-D unsteady state numerical examples, studying 1-D cases
can understand how the general time response temperature gets by distributed transfer
function method (DTFM). Comparing with traditional method, separation of variables,
the proposed method (DTFM) doesn’t need to compute two temperature
distribution and separately (i.e.,
1
() Tx
2
(, ) Txt
12
(, ) ( ) (, ) Txt T x T xt = + ) while the boundary
105
conditions are non-homogeneous. The separation of variables method needs to compute
“integral by part” on the initial conditions (i.e.,
0
0
l
l
u vdx u v uv dx
0
l
′ ′ =−
∫∫
), but the DTFM
gets the identical solution as separation of variables without this procedure. The DTFM
method applies arbitrary boundary conditions easily. From (4.36~4.39) and (2.39), the
temperature in the Laplace domain can be known easily. In heat conduction analysis, the
general time response temperature is in the time domain, so finding an inverse Laplace
solution is quite important. First of all, the procedure begins with the convolution integral.

00
ˆˆ ˆˆ
(, ) [ (, , ) ( , ) ( , ) ( )] , (0, )
tL
xtGxtQ dHxtRdx ηξτξτξ τττ =− +− ∈
∫∫
L (4.44)
And then, it applies the residue theorem. According to Yang’s paper (1996), the Green’s
function in (4.44) states as following
(4.45)
1
ˆ
(, ) Res[ (, )]
k
t
k
Hxt e Hx s
μ
∞
−
=
=
∑
The Laplace domain is ( , ) Hxs
() () 1
(() () )
Fs x Fs L
eMs Nse
−
+ and ( , , ) Gx s ξ contains
based on equation (2.39).
( , ) Hxs
If and
()
0,
n
Fx
e
μ −
≠
()
adj( ) 0
n
FL
MNe
μ −
+=
()
det( ) 0
n
FL
n
d
MNe
d
μ
μ
−
−
+ ≠ , and
n
μ is a simple
pole of the based on the basic residue theorem, then the residue value is ( , ) Hxs

() () () ()
()
()
adj( ) adj( )
Res
det( )
det( )
nn nn
n
n
n
Fx F L Fx F L
FL
s
FL
n
eMNe eMNe
d
MNe
MNe
d
μμ μμ
μ
μ
μ
μ
−− −−
−
=−
−
+
=
−
+
+
+
. (4.46)
The no heat generation in heat conduction governing equations with different boundary
conditions and zero initial condition state,
106

2
2
00
1
;( ,0) 0
(0, )
(0, )
(,)
(,)
LL
TT
Tx
tx
Tt
Tt
x
TLt
TLt
0
L
x
α
φ ϕ λ
φ ϕλ
∂∂
==
∂∂
∂
+
∂
∂
=
+ =
∂
(4.47)
Applying equations (4.34~4.39), the matrices are: , , , , and FQ M N R

0 00
01
00 0
;; ; ;
00 0 0
LL L
MN R F Q
λ φϕ
μ
φϕ λ
α
⎡⎤
⎡⎤ ⎡⎤ ⎡⎤ ⎡⎤
=
⎥
⎦
⎢⎥
== = =
⎢⎥ ⎢⎥ ⎢⎥ ⎢
⎢⎥
−
⎣⎦ ⎣ ⎣⎦ ⎣⎦
⎣⎦
s ; μ =− (4.48)
The eigenvalues are from
()
det( ) 0
n
FL
MNe
μ −
+ = .
()
00 0 0
det( )
1
sin cos cos sin 0
n
FL
nn n nn
LL L L
n
MNe
LL L
μ
μμ μ μμ
φ φ φ ϕ ϕφ ϕϕ
αα α αα μ
α
−
+
⎡⎤
⎢⎥
⎢⎥
=+ − +
⎢⎥
⎢⎥
⎣⎦
L=
(4.49)
The general time response temperature from equations (4.44~4.46) is

0
1
(, ) [ ]
n
t
n
Txt e d
μτ
τ
∞
−
=
Ξ
=×
Θ
∑
∫
(4.50)

00 0
0
1
sin ( ) cos ( ) cos
1
sin
nn
LL
n
n
L
n
n
L
xLxL
x
x
μ μμ
φλ ϕ λ ϕ λ
α αα μ
α
μ
φλ
α μ
α
Ξ= − − + − −
+
(4.51)
107
()
0
00 0
1
sin cos
2
2
1
sin sin cos
2
22
nn
L
n n
n
nn
LL L
nn
L
LL
LL
LL
μμ
φφ
αμ α μ
μ
α
μμ
φ ϕ ϕφ ϕϕ
ααα μμ
αα
αα
⎡⎤ ⎛⎞
⎢⎥ ⎜⎟
⎢⎥ ⎜⎟
Θ= −
⎢⎥ ⎜⎟
⎜⎟ ⎢⎥
⎝⎠ ⎣⎦
n
L
μ
α
⎡ ⎤ ⎡⎤⎛ ⎞
⎢ ⎥ ⎜⎟ ⎢⎥
−−
⎢ ⎥ ⎜⎟ ⎢⎥
+− + +
⎢ ⎥ ⎜⎟ ⎢⎥
⎜⎟ ⎢ ⎥ ⎢⎥
⎣⎦⎝ ⎠ ⎣ ⎦
(4.52)

Example 4.6

2
2
1(,)
;(0,) 0; ; ( ,0) 0
TT TLt
Tt k qTx
tx x α
∂∂ ∂
== =
∂∂ ∂
=
The eigenvalues from equation (4.49) are

2
()
(2 1) (2 1)
det( ) cos 0;
22
n
FL nn
n
nn
MNe L L
L
μ
μμ π π
μα
αα
−
++ ⎛⎞
+== = ⇒=
⎜⎟
⎝⎠
.
The general time response temperature from equations (4.50~4.52) is

0
1
1
sin
(, ) [ ]
sin
2
n
n
n
t
n
n
n
q
x
k
Txt e d
L
L
μτ
μ
α μ
α
τ
μ
α μ
α
α
∞
−
=
=×
∑
∫

2
(2 1)
2
22
1
(2 1)
8(1)sin
2
(2 1)
n
n
t
L
n
n
Lq x
q
L
xe
kkn
π
α
π
π
+ ⎛⎞
∞
−
⎜⎟
⎝⎠
=
+
−−
=+
+
∑
.
The solution is exactly identical to the separation of variable method. They have the same
eigenvalues and same general time response temperature distribution.
108
Example 4.7

2
2
1(0,) (,)
;0; (,);(,
L
TT T t TLt
kk hTLtTTx
tx x x α
∂∂ ∂ ∂
=− = + = =
∂∂ ∂ ∂
0)0
=

The eigenvalues from equation (4.49) are
.
()
det( ) cos sin 0
n
FL
nn n
MNe h L k L
μ
λλ λ
−
+=− +
The general time response temperature from equations (4.50~4.52) is

0
1
cos
(, ) [ ]
sin sin cos
222
n
n
L
t
n
nn n
nn
Tx
Txt e d
hL k kL
LL L
μτ
μ
α
τ
λλ λ
αλ αλ α
∞
−
=
=×
++
∑
∫

2
1
sin
cos
sin 2
24
n
n L
t n L
n
n n
n
L T
h T
xe
L L
h
α λ
λ
λ
λ
λ
λ
∞
−
=
−
=+ ×
+
∑
.
The solution is exactly identical to the separation of variable method. The notation
n
μ is
equal to the thermal diffusivity of the material α multiples the square of eigenvalues
n
λ .
They have the same eigenvalues and same general time response temperature distribution.
The author finds an interesting phenomenon, the Fourier series effect. Take a close look
on the two previous examples.
In the examples 4.6, the steady state term
22
1
(2 1)
8(1)sin
2
(2 1)
n
n
n
Lq x
q
L
x
kkn
π
π
∞
=
+
−
=
+
∑
with
1
(2 1)
sin
2
n
n
qn
xax
kL
π
∞
=
+
=
∑
and
0
2(21)
sin
2
L
n
qn
ax
Lk L
xdx
π +
=
∫
.
109
In the example 4.7, the steady state term
1
sin
cos
sin 2
24
L
n L
n
n n
n
n
T
L T
h
x
L L
h
λ
λ
λ
λ
λ
∞
=
=
+
∑
with
1
cos
L
nn
n
T
ax
h
λ
∞
=
=
∑
and
0
2
0
cos
cos
L
L
n
n L
n
T
xdx
h
a
xdx
λ
λ
=
∫
∫
.
The following figure provides two kinds of solutions.
Solution (1) is
2
(2 1)
2
22
1
(2 1)
8(1)sin
2
(2 1)
n
n
t
L
n
n
Lq x
q
L
xe
kkn
π
α
π
π
+ ⎛⎞
∞
−
⎜⎟
⎝⎠
=
+
−−
+
+
∑
.
Solution (2) is
22
1
(2 1)
8(1)sin
2
(2 1)
n
n
n
Lq x
L
kn
π
π
∞
=
+
−
+
∑
2
(2 1)
2
22
1
(2 1)
8(1)sin
2
(2 1)
n
n
t
L
n
n
Lq x
L
e
kn
π
α
π
π
+ ⎛⎞
∞
−
⎜
⎝
=
⎟
⎠
+
−−
+
+
∑
.

0 0.1 0.2 0.3 0.4 0.5 0.6 0.7 0.8 0.9 1
0
0.5
1
1.5
2
2.5
x
Temperature
T F (1-D Heat Conduction-Cartesian) Fourier series effect

solution-(1)
solution-(2) 1 term
solution-(2) 5 terms
solution-(2) 15 terms

Figure 4.14 the Fourier series effect for example 4.6
110
The example 4.6 is one-dimensional heat conduction with boundary condition TF in the
Cartesian coordinate systems. The prescribed temperature T along left edge assumes .
The heat flux
0
0 C
flux
q sets .
2
100 / Wm
Although both solution (1) and solution (2) are the same meaning, it is still little
difference between Fourier series steady state term and non-Fourier series steady state
term in the computer programming. Fourier series term needs infinite terms to
approximate the non-Fourier series term. The curve with the Figure with Fourier series
steady state term solution looks like a wave function. The curve with non-Fourier series
term steady state solution is smoother. This important investigation leads an important
concept.
The general time domain solutions by the finite difference distributed transfer function
(FDDTFM) are given in equation (2.43). The time response solution contains two parts:
steady state part and unsteady state (transient) part. Since the steady state part in the time
response solution from the FDDTFM has wave-like curve, this part should throw away.
One question remains: how to get the smoother curve in the steady state part? The
dissertation provides an amazing answer for this issue. The previous half part derivation
on each chapter, there is a static/steady state analysis. The final time response solutions
are complete with that static/steady state analyses part in the static/steady state solution
and the transient/unsteady state part in the time response solution.

section 3.8;4.8;5.7 section 3.9;4.9;5.8
Time response Steady State part Transient part =+

(4.53)
This technique makes the curve smoother.

111
Example 4.8
The prescribed temperature T along left edge assumes . The prescribed temperature
along bottom edge assumes . The prescribed temperature T along top edge
assumes . The heat flux
0
0 C
T
0
0 C
0
0 C
flux
q along the right edge sets .
2
100 / Wm

Figure 4.15 the square heat conduction with TTFT boundary conditions
The time response solutions for temperature and flux are shown in Figure 4.16 ~4.19.
The FDDTFM convergence is guaranteed. (Figure 4.16 & 4.18) The FDDTFM uses 15
mesh lines along the direction with 34 eigenvalues and 25 sec to get the closer steady
state solution than analytic solution. The time period is from 1 sec to 25 sec, the
convergences are guaranteed in the temperature and flux distribution. The analytical
solution doesn’t match boundary conditions well due to the how many terms compute
together. The boundary condition of the FDDTFM guarantee along left edge temperature
is zero and along right edge heat flux is 100 .
y
2
/ Wm
112

0 0.1 0.2 0.3 0.4 0.5 0.6 0.7 0.8 0.9 1
-0.2
0
0.2
0.4
0.6
0.8
1
1.2
x
Temperature
T T F T (2D Heat Conduction-Cartesian)

FDDTFM (time response) 34 eigenvalues; 25sec; 3 lines
FDDTFM (time response) 34 eigenvalues; 25 sec; 7 lines
FDDTFM (time response) 34 eigenvalues; 25 sec; 15 lines
Analytical (time response) 35 eigenvalues; 25 sec

Figure 4.16 distribution of general temp. on A-B for the TTF region by mesh lines T

0 0.1 0.2 0.3 0.4 0.5 0.6 0.7 0.8 0.9 1
0
0.1
0.2
0.3
0.4
0.5
0.6
0.7
0.8
0.9
1
x
Temperature
T T F T (2D Heat Conduction-Cartesian)

FDDTFM (time response) 15 lines; 34 eigenvalues; 1 sec
FDDTFM (time response) 15 lines; 34 eigenvalues; 10 sec
FDDTFM (time response) 15 lines; 34 eigenvalues; 25 sec
FDDTFM (steady) 15 lines

gure 4.17 distribution of general temperature on A-B for the TTFT region by tim Fi e
113

0 0.1 0.2 0.3 0.4 0.5 0.6 0.7 0.8 0.9 1
0
20
40
60
80
100
120
x
Flux
T T F T (2D Heat Conduction-Cartesian)

FDDTFM (time response) 34 eigenvalues; 25sec; 3 lines
FDDTFM (time response) 34 eigenvalues; 25 sec; 7 lines
FDDTFM (time response) 34 eigenvalues; 25 sec; 15 lines
Analytical (time response) 35 eigenvalues; 25 sec

gure 4.18 distribution of general flux on A-B for the TTFT region by mesh line Fi s

0 0.1 0.2 0.3 0.4 0.5 0.6 0.7 0.8 0.9 1
0
20
40
60
80
100
120
x
Flux
T T F T (2D Heat Conduction-Cartesian)

FDDTFM (transient) 15 lines; 34 eigenvalues; 1 sec
FDDTFM (transient) 15 lines; 34 eigenvalues; 10 sec
FDDTFM (transient) 15 lines; 34 eigenvalues; 25 sec
FDDTFM (steady) 15 lines

gure 4.19 distribution of general flux on A-B for the TTFT region by tim Fi e
114
The concept of equation (4.53) is very important here. The FDDTFM convergence is
guaranteed. Those figures contain the steady state response and time response. The
analytical solutions are Fourier series, so the boundary condition is not exactly equal to
100 in the Figure 4.18.
2
/ Wm

Example 4.9
The choice of material I on the left side is carbon steel 1.0%. The thermal
conduction 43 /( )
o
Wm C ⋅ . The density of the bo k = dy ρ =7801
3
/ kg m . The specific heat
of material c =0.473
0
/( ) kJ kg C ⋅ . The choice of material II on the right side is Aluminum
bronze (95% Cu; 5% Al). The thermal conductio )
o
C n =83W k /(m ⋅ . The density of the
body ρ =8666
3
. The specific heat of materia c =0.41 ) . (Holman, 1986 / kg m l 0
0
/( kJ kg C ⋅ )
The prescribed temperature T along left edge assumes . The prescribed temperature
along bottom edge assumes . The prescribed temperature T along top edge
assumes . The heat flux
0
0 C
T
0
0 C
0
0 C
flux
along the right edge s
2
m . The height H 1 m .
The length 1 L is equal to 0.7 m . The length 2 L is equal to 0.3
q et100 / W is
.

s
m
From Equation (2.20) with perfect thermo contact assumption, the connection matrix is C
0
0
I
II
I
C k
I
k
⎡
⎢
=
⎢⎥
⎢⎥
⎣⎦
⎤
⎥
. The identity matrix I is nn × matrices. The mash lines are n .
At point
1
x , the perfect thermo contact has relationship:
1
()
II
x η is equal to
1
()
I
Cx η .
The relationship between
2
x and
1
x (Yang, B., & Fang, H., 1994) is

II I
22 1 1
20
() ( )
FL FL
x eCe x ηη = . (4.54)
115
The state variables vectors
0
()
I
x η are derived from boundary condition equation in
equation (4.38).

22 1 1
1
)
FL FL
0
() ( x
I
MN η =+ e Ce R
−
(4.55)

re 4.20 the composite heat conduction with TTTFTT boundary conditio Figu ns

The general time response solutions for temperature and flux are shown in Figure 4.21
~4.24. The FDDTFM convergence is guaranteed. (Figure 4.21 & 4.23) The FDDTFM
uses 13 mesh lines along the y direction with 29 eigenvalues and 20 sec to get the closer
steady state solution than FEM (2500 mesh elements) solution. The time period is from 1
sec to 20 sec, the convergences are guaranteed in the temperature and flux distribution.
The boundary condition of the FDDTFM guarantee along left edge temperature is zero
and along right edge flux is 100 / Wm
2
.

116

0 0.1 0.2 0.3 0.4 0.5 0.6 0.7 0.8 0.9 1
-0.5
0
0.5
1
x
Temperature
T T T F T T(2D Heat Conduction-Cartesian) 2 layers

FDDTFM (time response) 29 eigenvalues; 20sec; 3 lines
FDDTFM (time response) 29 eigenvalues; 20 sec; 7 lines
FDDTFM (time response) 29 eigenvalues; 20 sec; 13 lines
FEM (time response) 50X50 elements; 20 sec

Figure 4.21 distribution of general temp. on A-B for the TTFTT region by mesh lines T

0 0.1 0.2 0.3 0.4 0.5 0.6 0.7 0.8 0.9 1
-0.5
0
0.5
1
x
Temperature
T T T F T T(2D Heat Conduction-Cartesian) 2 layers

FDDTFM (time response) 13 lines; 29 eigenvalues; 1 sec
FDDTFM (time response) 13 lines; 29 eigenvalues; 8 sec
FDDTFM (time response) 13 lines; 29 eigenvalues; 20 sec
FDDTFM (steady) 13 lines

re 4.22 distribution of general temperature on A-B for the TTTFTT region by ti Figu me
117

0 0.1 0.2 0.3 0.4 0.5 0.6 0.7 0.8 0.9 1
-20
0
20
40
60
80
100
120
x
Flux
T T T F T T(2D Heat Conduction-Cartesian) 2 layers

FDDTFM (time response) 29 eigenvalues; 20sec; 3 lines
FDDTFM (time response) 29 eigenvalues; 20 sec; 7 lines
FDDTFM (time response) 29 eigenvalues; 20 sec; 13 lines
FEM (time response) 50X50 elements; 20 sec

re 4.23 distribution of general flux on A-B for the TTTFTT region by mesh lin Figu es

0 0.1 0.2 0.3 0.4 0.5 0.6 0.7 0.8 0.9 1
-20
0
20
40
60
80
100
120
x
Flux
T T T F T T(2D Heat Conduction-Cartesian) 2 layers

FDDTFM (time response) 13 lines; 29 eigenvalues; 1 sec
FDDTFM (time response) 13 lines; 29 eigenvalues; 8 sec
FDDTFM (time response) 13 lines; 29 eigenvalues; 20 sec
FDDTFM (steady) 13 lines

re 4.24 distribution of general flux on A-B for the TTTFTT region by ti Figu me
118
4.10 Discussion
119
ge w
In the example 4.1, it is easy to check the left edge boundary condition with
temperature and right edge boundary condition with temperature
0
200 Cin Figure
4.5. The absolute value of difference between FDDTFM and FEM divided by FEM
defines as an error (difference). In the example 4.2, it is straightforward to check
boundary condition on the left ed ith temperature
0
20 C and boundary on the right
edge with adiabatic boundary condition (slope 0
0
20 C
= ). (i.e., Figure 4.7). In the example 4.3,
the boundary condition on the
0
0 edges is temperature
0
200 C . The boundary condition
on the
0
90 edges is temperature
0
20 C . (i.e., Figure 4.9) In the example 4.4, the boundary
ndition on the
0
0 edges is adiabatic boundary condition (slope 0 = ). The boundary
dition on the
0
90 edges is temperature
0
20 C . This situation displays in Figure 4.11.
From now on, the investigation on the boundary condition can provide confirmative
solution case by case. In the example 4.5, the boundary condition at the G corner is
adiabatic boundary condition (slope 0
co
con
= ). The boundary condition at the D corner is
temperature
0
100 C . (i.e., Figure 4.13) In the example 4.6 and 4.7, the Distributed
Transfer Function Method (DTFM) in 1-D on heat conduction problems gets the exactly
identical solution to the analytical solution. In the Figure 4.14, the Fourier series effect
discussion leads a very important concept shown in (4.53). The example 4.8 shows very
good time response approximation on the 2-D heat conduction with TTFT boundary
conditions. The boundary condition can easily verify in FDDTFM. (i.e., Figure 4.16
~4.19) The example 4.9 shows the FDDTFM has better time response approximation
than FEM on the 2-D 2-layer composite heat conduction with TTTFTT boundary
120
uarantees the boundary condition on the edge match well.
conditions. The boundary condition can easily verify in FDDTFM. (i.e., Figure 4.21
~4.24)The FDDTFM gets solution easily no matter how boundary conditions change.
The proposed method g

Chapter 5

Static and Dynamic Analysis of Plate Problems

5.1 Problem Statement
This dissertation gives a novel method called finite difference distributed transfer
function method (FDDTFM) to solve the plate problems. The plate governing equation
(2.14) derived from constitutive (2.2), kinematics (2.8), resultant (2.9; 2.10), and
equilibrium equations (2.11; 2.12; 2.13) under Kirchhoff-Love hypotheses. The proposed
method uses three major boundary conditions (2.21; 2.22; 2.23) to solve rectangular and
circular plate problems.

5.2 Proposed Method in the Cartesian coordinate Systems
5.2.1 Finite difference discretization in the Cartesian coordinate systems
The current case contains n mesh points; n mesh lines (Fig. 2.1). The deflection of the
plate on mesh line located at( , )
th
k x kh defines
k
w . The subdivision height between two
mesh points is h . The total height of the plate is H . The total width of the plate is L .
(, ) (, );
k
H
w w xk y w xkh y h
k
=Δ= Δ== (5.1)
The deflection of the plate on the x direction is . The Finite difference method along y
direction applies on the plate governing equations in the Cartesian coordinate systems
k
w
121
(2.14). The Central finite difference scheme for plate governing equations in the
Cartesian coordinate systems is introduced.

( )
21 1 2 11
24
46 4 24 2
kk k k k kk k t
k z
ww w w w ww w h
ww
hh
ρ
−− + + −+
−+ −+ ′′ ′′ ′′ −+
′′′′++ + p
D
=
means
(5.2)
5.2.2 Boundary conditions in the Cartesian coordinate systems
In this dissertation, there are four-side boundary conditions for plate problems.
The proposed method concerns the OB & the AD sides first in Fig. 5.1. After that, it
regards as the other two opposite sides, the OA & the BD sides. The notation
CSCF that edge 0 ( ) x i is clamped ( ) C on the edge. The edge 0 ) = ( y ii = is
simple supported( ) S . The edge ( ) x Liii = is clamped ( ) C . The edge ( ) y Hiv = is
free ( ) F . The paper is going to i te the middle dotted line( ) m "" .The detail
information of the mesh lines and the Cartesian coordinate compo
nvestiga
nent detailed
formation is in Figure 2.1. in

lar plate with boundary conditions Figure 5.1 an example of rectangu

122
The first part considerations want to make form such as ( ) (0, ) ( ) ( , ) ( ) Mtt Nt Lt Rt η η= .
(,) [ (,) ( ,), ,) (,),
),
nn
mt wmtwmtwmtwmt
wm
+

(
123
11
′ η
11
( , ) ( , ( , ) ( , )]; 0 or
nn
wmt w mt t w mt m L
′ =
′′ ′′ ′′′ ′′′
""
(5.3)
5.2.2.1.1 The line
= ""
5.2.2.1 Left nodes
with clamped boundary condition 0 ( ) x i =

00 0 0
;
x lx l
ww
w
x
θ
==
∂
==
∂
(5.4)
he deflection of the plate i The initial d
f the plate i
T eflection of the plate is
0l
w ; the initial angle s w .
. For the finite mesh points,
l 00
and
kl k
ww w θ ′ = =. o s
0l
θ
5.2.2.1.2 The line 0 ( ) x i = with simple supported boundary condition

22
00 0 0 0
22
;
xlx x
ww
wwM D M
xy
ν
==
⎛⎞ ∂∂
==− + =
⎜⎟
∂∂
⎝⎠

l=
(5.5)
The initial bending moment of the plate is
0l
M . The first derivative and second derivative
of w with respect to y are zero, because the first condition
00
(constant)
xl
ww
=
= is
function of x only. (i.e.,
2
0
kk
ww ∂∂
2
yy
= = ) For finite mesh points, ww = and
0 kl
0l
∂∂
k
M
w ′′ =− .
0 (
D
5.2.2.1.3 The line ) x i = with free edge boundary condition
22 3 3

00 0 0 00
22 3 2
;(2)
x xl x x l
ww w w
DMD V νν
== = =
⎛⎞ ⎛ ⎞ ∂∂ ∂ ∂
−+ =− +− =
⎜⎟ ⎜ ⎟
(5.6)

xy x xy ∂∂ ∂ ∂∂
⎝⎠ ⎝ ⎠
22
2200
11 1 1
(2 ) ; (2)( 2 )
ll
k k kk k k kk
MhVh
hw vw ww hw vw ww
DD
−+ −+
′′ ′′′ ′ ′ ′ +− + =− +− − + =− (5.7)
The initial effective shear force is . The detaile ear force described in
0l
V d effective sh
section 2.3.3.3.
5.2.2.2 Right nodes
5.2.2.2.1 The procedure on line () x L iii = is similar with the left nodes
cheme throughout the whole
issertation. This technique avoids the ex
), there are several boundary conditions. On the first
h ts
edge mesh lines (
This current case uses the central finite difference s
d tra unknown terms. On the mesh lines
(0
th
0 ( ) yii = and
th
n () yH = iv
mesh line, it contains the -1
st
and the 0
th
mesh points. On the n
th
mesh line, it contains the
n+1
th
and the n+2
th
mes poin . Different boundary conditions cause different
representations. This dissertation shows three type boundary conditions along these two
special edge mesh lines. The following subsections will show how to deal with the
different type boundary conditions on the 0 ( ) y ii = and ( ) y Hiv = ).
here are two catalogues. One is on the 1
st
mesh line and the other one is on the n
th
mesh
e has thr
he successive discussion will make form lik

T
line. Each mesh lin ee type boundary conditions.
T e following,
[ ] [ ] [ ] [ ] [ ]
[ ] [ ] [ ] [ ] []
1
4
1, ,1 1,2 2
t
k k k k k k k k k k z
hh
C w C w C w w P
D
ρ
+
− + + + +
+ + + + =
(5.8)
.2.2.3 Bottom nodal lines
inite difference scheme.
inally, important terms and
,,11 ,, 1 ,22
,1
kk k kk k kk k kk k k k k
kk k
Aw B w B w B w C w
Cw
−− + − −
−
′′′′ ′′ ′′ ′′ ++ + +
+
5
Follow the plate boundary conditions definition and use central f
0
w
1
w
−
124
F gets from the fundamental c
important terms substitute into the 1
st
mesh line of the plate governing equation. This
procedure can remove the extra finite difference unknown terms, such as and
omputation. These
0
w
1
w
−
in
1
st
mesh line of the plate governing equation.

5.2.2.3.1 The line 0 ( ) y ii = with clamped boundary condition
11
;
kk
ww w
ww w
00 0 0
2
yk b y b
yh
θ
+−
∂−
== =
The initial deflection o e plate is w ; the initial slop
==
=
∂
(5.9)
f the plate of th e of the plate is
0b 0b
θ .
For finite mesh points,

00 1 1 0
and 2
bb
ww w w h θ
−
= =−. (5.10)
.2.2.3.2 The lin i with simple supp

5 orted boundary condition e 0 ( ) yi =
22
ww ⎛⎞ ∂∂
00 0 0 0
22
;
yb y y b
wwM D M
y
ν
===
==− + =
⎜⎟
∂
⎝
(5.11)
he initial bending moment of the plate is
x∂
⎠
0b
M . The first derivative and second derivative T
00
(constant)
yb
ww
=
= of wwith respect to xare zero, because the first condition here is
125
function of y only. (i.e.,
2
2
0
xx
kk
ww ∂∂
= =
∂∂
). For the finite mesh points,

2
0
;2
b
00 1 1 0 bb
M h
ww w w w ==−+ − . (5.12)
D
−
5.2.2.3.3 The line i with free edge boundary condition 0 ( ) yi =
22 3 3
0 0 0 0
22 3
;(2)
y b y
ww w
DMD
yx y y
= =
−+ =− +−
⎜⎟ ⎜
∂∂ ∂ ∂
⎝⎠ ⎝
00
2
y y b
w
V
x
νν
= =
⎛⎞ ⎛ ⎞ ∂∂ ∂ ∂
=
⎟
∂
⎠
(5.13)
ffective shear force is . For the finite m

The initial e esh points,
0b
V
2
2 0
21 0 1
3
2

0
0
)
b
D
′′=−
32 0 1 2
)
(2 2 )(2 )(
b
Mh
ww w hvw
D
Vh
ww w w vhw w
−
′′ −+ + =−
′′ −+ − + − −
. (5.14)
(2

126
5.2.2.4 Upper nodal lines
Follow the plate boundary conditions definition and use central finite difference scheme.
Finally, important terms
1 n
w
+
and
2 n
w
+
gets from the fundamental computation. These
portant terms substitute into the n
th
mesh line of th quation. This im e plate governing e
procedure can remove the extra finite difference unknown terms, such as and
1 n
w
+ 2 n
w
+
in
n
th
mesh line of the plate governing equation.
5.2.2.4.1 The line () y Hiv = with clamped boundary condition
11
00 0
;
2
kk
yk t y
ww w
ww w
yh
0t
θ
+−
==
∂−
===
∂
= (5.15)
e of the plate is The initial deflection of the plate of the plate is ; the initial slop
0t
w
0t
θ . For
esh points,
0t
finite m
10 2
; 2
ntn n
www w h θ
++
= =+. (5.16)
() yH iv = 5.2.2.4.2 The line with simple supported boundary condition
22
00 0 0
22
;
yt y y
wwM D M
yx
ν
===
==− + =
⎜⎟
∂∂
⎝⎠
0t
ww ⎛⎞ ∂∂
(5.17)
The initial bending moment of the plate is
0t
M . The first derivative and second derivative
00
(constant) ww = is function of w with respect to x are ze o, because the first condition r
yt =
of y only. (i.e.,
2
2
xx ∂∂
0
kk
ww
== ). For the finite mesh points,
∂∂

2
0
2
t
10 2 0
;
ntn n t
M h
w w + − . (5.18) www
++
==−

D
5.2.2.4.3 The line () y Hiv = with free edge boundary condition
2
w⎛∂
2 3 3
00 0
22
;(2)
yy t
w w w
DMD V
yx
νν
==
⎞ ⎛ ⎞ ∂ ∂ ∂
−+ =− +− =
⎜⎟ ⎜ ⎟
∂∂
⎝⎠
(5.19)
s,

0 00
3 2
y y t
y yx
= =
∂ ∂∂
⎝ ⎠
The initial effective shear force is
0t
V . For the finite mesh point

2
2 0
11
3
21 1 2 1 1
(2 2 )(2)( )
t
nn n n
nn n n n n
Mh
D
ww w w vhw w
D
+−
++ − − + −
′′ ′′ −+ − +− − =−
. (5.20)

5.3 Proposed Method in the polar coordinate Systems
5.3.1 Finite difference discretization in the polar coordinate systems
The curren c
2 0
(2 )
t
www hvw
Vh
′′ −+ + =−
t ase contains mesh points; mesh lines (Fig. 2.2). The deflection of the
plate on mesh line located at
n n
th
k ( , ) kh θ defines . The subdivision height between two
mesh points is . The total difference between inner radius and outer radius is .

k
w
h H
(, ) ( , );
k
H
wwkr wkh r h
k
θθ =Δ = Δ= = (5.21)
The deflection of the plate at ( , ) kh θ is . The Finite difference method along
direction applies on the plate governing equations in the polar coordinate systems (2.29).
The Central finite difference scheme for plate governing equations in the polar coordinate
stems is introduced.

k
w r
sy
21 1 2 2 1 1 2
3
11 11 1 1
22 3 2 2
11
344
2
12 1 2 2
()
2
241
()
2
nn n n n n n n n
nn n n n n n n
nn t
nn
h
www w w www
rh r h r h
ww h p
ww w
rh r r D D
ρ
− − ++ − − ++
−+ −+ −+
−+
+
′′ ′′ ′′ −+ − + −+
−+ +
′′ ′′ −+
′′ ′′′′ − +++ =
(5.22)
46 4 2 22 ww w w w w w w w −+ −+ − + −+
4
hr
127
128
.3.2 Boundary conditions in the polar coordinate systems
this dissertation, there are four-side boundary conditions for the plate problems. The
erns the OB & the ED sides first in Fig. 5.2. After that, it regards
sides. The notation means
at edge
5
In
proposed method conc
CSCF as the other two opposite sides, the arc OE& the arc BD
()
2
i
π
θ = is clamped . The edge ( ) C
0
() rrii = is simple supported( ) S . The th
edge 0( ) iii θ = is clamped( ) C . The edge
0
() rr Hiv = + is free edg ) . The detailed
information of the mesh lines and the pola t is in Figure 2.2. The
e F
omponen

(
r coordinate c
dissertation is going to investigate the middle dotted line() m "" .

Figure 5.2 an example of annular sector plate with boundary conditions
he first part considerations want to make form such as T ( ) M ( ) (0, ) ( ) ( , ) tt Nt t Rt η η +Θ
11
(,) [ (,) ( ,),
( , ) ( , ),
n
nn
mt w m t w mt w
wmtwmtw
= .

1
( ,) (,),
( , ) ( , )]; 0 or
n
m t w mt
mtwmt m
′
1
′ =
′′ ′′
η
′′′ ′′′ = Θ
(5.23)
""
""

5.3.2.1 Right nodes
5.3.2.1.1 The line with clamped boundary condition 0 ( ) iii θ =
00
;
kr k
w
ww w
129

r θ θ
θ
∂
θ
′ ===
∂
(5.24)
The deflection of the plate is w . The initial deflection of the plate is
0 r
w
θ
; the initial angle
of the plate is
0 r θ
θ . For the finite mesh points,
00
and
kk
kr k r
ww w
θ θ
θ ′ = =.
5.3.2.1.2 The 0 ( ) iii θ = simple supported boundary condition

line with
22
11
;
ww w
ww Dv M
rrr r
00
222
kr r θ θ ⎢⎥ ⎜⎟
θ
⎡⎤ ⎛⎞ ∂∂ ∂
=− + + =
∂∂ ∂
⎝⎠ ⎣⎦
(5.25)
The first derivative and second derivative of w with respect to r are zero, because the
first condition
0
(constant)
kr
ww
θ
= is function of θ only (i.e.,
22
0 wr w r ∂ ∂=∂ ∂ = ).For
finite meshes points,
2
00
and /
kk
kr k r
ww w rM D
θθ
′′ ==− .
ing moment of the plate is
0
The initial bend M
θ
.
5.3.2.1.3 The 0 ( ) iii θ = with free edge boundary condition

22
11 ww w
0
22
r
M
r
2
Dv
rrr
θ
θ
+ =
⎥⎟
∂
⎠⎦
(5.26a)
⎡⎤ ⎛⎞ ∂∂ ∂
−+
⎢ ⎜
∂∂
⎝⎣

2
11 w
2 2
1 1 1
(1 )
w w w w
Dv V
0
222 2
r
rr rrr rrr r
θ
θθ θθ
⎢⎥ ⎜⎟ ⎜ ⎟
∂∂ ∂ ∂ ∂ ∂∂ ∂
⎝⎠ ⎝ ⎠ ⎣⎦
⎡⎛ ∂∂
⎤ ⎞ ⎛ ⎞ ∂ ∂ ∂ ∂ ∂
−++ +− − = (5.26b)

22
0
2
r
22 22
11
2(2 2) 4 (2 2)
k
kk k k
M rh
D
θ

hw vr rh w vr w vr rh w
−+
′′+− − ++ =− (5.27a)
22 2 2 2 2 2
1
1
2(4 2 2) (8 4 4 4)
(4 2 2 )
kk
k
hw r r v r rv w r r v h h v w
rrvr rvw
−
+
′′′ ′ +− +− +− + + −
+− −+ =−
0 22 32
2
k
k
r
V
rh
D
θ
′
′
(5.27b)
130
ve shear The initial effecti force is
0
V
θ
. The detailed effective shear force described in
ction 2.3.3.3.
ures on
se
5.3.2.2 Left nodes
5.3.2.2.1 The proced /2 ( ) i θ π = is similar with the left nodes
his current case uses the central finite difference sch
s. On the mesh lines
and ), there are several boundary conditions. On the first
mesh line, it contains the -1
th
mesh points. On the n
th
mesh line, it contains the
o
t e w three type boundary conditions along these two
deal with the
ifferent type boundary conditions on t
). There are two catalogues. One is on the 1
st
one is on the n
th
mesh line. Each mesh line has three type boundary conditions.
The successive discussion will make form like following,

T eme throughout the whole
dissertation. This technique avoids the extra unknown term
(0
th
0
() rr ii =
th
n
0
() rr H iv =+
st
and the 0
n+1
th
and the n+2
th
mesh points. Different boundary c nditions cause different
representa ions. This diss rtation sho s
special edge mesh lines. The following subsections will show how to
d he edge nodal lines ( 0
th
0
() rr ii = and
th
n
0
() rr H iv =+ mesh line and the other
[ ] [ ] [ ] [ ] [ ]
[] [ ] [] [ ] []
,,11 ,,11 ,2
44
,1 1 , , 1 1
2
kk kkk k kk kkk k kk
t
kk k kk k kk k
Aw B w B w B w C
hr h
Cw C w C w C w w P
ρ
−− + + −
−− + +
′′′′ ′′ ′′ ′′ ++ + +
++ + + + =
(5.28)
ollow the plate boundary conditions definition and use central finite difference s
Finally, important terms and
2k
w
−
, 2 2 kk k k z
D
+ +
5.3.2.3 Bottom nodal lines
F cheme.
0
w
1
w
−
gets from the fundamental computation. These
portant terms substitute into the 1
st
mesh line of the plate governing equati im on. This
131
extra finite difference unknown terms, such as and
0
w
1
w
−
procedure can remove the in

1
st
mesh line of the plate governing equation.
5.3.2.3.1 The arc
0
() rr ii = with clamped boundary condition
11
00
;
2
kk
ww w
rh
krb rb
ww θ
+−
∂ −
== =
∂
(5.29)
; the initial slope of the plate is The initial deflection of the plate of the plate is
0rb
w
0rb
θ .
For finite mesh points,

0 0
00 1 1 0
and 2
rb rb
ww w w h θ
−
= =−. (5.30)
5.3.2.3.2 The arc with simple supported boundary condition
0
() rr ii =
22
00
222
11
,
krb
ww w
ww D v M
rrrr θ
⎡⎤ ⎛⎞ ∂∂ ∂
=− + + =
⎢⎥ ⎜⎟
∂∂ ∂
⎝⎠ ⎣⎦
rb
(5.31)
he first derivative and second derivative of

T w with respect to θ are zero, because the first
condition r only. (i.e.,
0
(constant)
krb
ww = is function of
22
ww θθ 0
kk
∂ ∂=∂ ∂ = )
The initial bending moment of the plate is
0b
M . For the finite mesh points,

0
00r
ww = (5.32a)
0
22
0

2
(2
2 2
10 1
2
2 ) 4 (2 2 )
r
M rh
vrhw rw r vrhw
D
−
− + + =− . (5.32b)
.3.2.3.3 The ar with free edge bounda
r−
5 ry condition c
0
() rr ii =
22
0
222
rb
M
rrrr θ
⎜⎟
∂ ∂
⎝⎠
(5.33a)
11 ww w
Dv
⎡⎤ ⎛⎞ ∂∂ ∂
−+ + =
⎢⎥
∂
⎣⎦

22 2
1 1 v w w ⎛ ⎞ − ∂ ∂ ∂
0
222 2
11 1
rb
ww w
DV
rr r r r r rr r θθ θ θ
⎡⎤ ⎛⎞ ∂∂ ∂ ∂
−++ + − =
⎢⎥ ⎜⎟⎜ ⎟
∂∂ ∂ ∂ ∂ ∂∂ ∂
⎝⎠⎝ ⎠ ⎣⎦
(5.33b)
132
The initial effective shear force is . For the finite mesh points,
0rb
V
0
22
0
22 2 2

010 1
2(2 2) 4 (2 2)
2
r
M rh
w
D
′′ =− (5.34a) vh w r vrh r w r vrh w
−
+− − + +
1
23 2 3 32
01 21

2
33
0 232 2 3
123
(2 ) 2 (3 ) (2 ) (2 2 )
2
4(2 2 )
r
rh v w h v w rh v w r w r r h rh w
Vrh
rhw r rh rh w r w
−
′′ ′′ ′′ −− − − + − − + + +
−+− + − + =−

es
Follow the plate boundary conditions definition and use central finite difference scheme.
Finally, important terms and
0
(5.34b)
D
5.3.2.4 Upper nodal lin
1 n
w
+ 2 n
w
+
gets from the fundamental computation. These
important terms substitute into the mesh line of the plate governing equation. This
procedure can remove the extra finite difference unknown terms, such as and
n
th
1 n 2 n
w
+
w
+
in
n
th
esh line of the plate governing equation. m
5.3.2.4.1 The arc
0
() rr H iv with clamped boundary condition =+

11
0
2
kk
rt
w
h
0
;
krt
ww
ww
r
θ
+−
∂ −
== =
∂
(5.35)
T of the plate is
0rt
he initial deflection of the plate of the plate is ; the initial slope
0rt
w θ .
For finite mesh points,

0t 10 2
; 2
ntn n
www w h θ
++
= =+. (5.36)
5.3.2.4.2 The arc with simple supported boundary condition

0
() rr H iv =+
22
00
222
krt rt
rrrr θ
⎢⎥ ⎜⎟
∂∂ ∂
⎝⎠ ⎣⎦
11
,
ww w
ww D v M
⎡⎤ ⎛⎞ ∂∂ ∂
=− + + = (5.37)
The initial bending moment of the plate is
0t
M . The first derivative and second derivative
of wwith respect
133
to θ are zero, because the first condition
0
(constant)
krt
ww = is function
f only. (i.e., r
2
2
0
kk
ww
θ θ
∂∂
==
∂ ∂
o ) For the fi esh points,

nite m
1
10
n
nr
ww
+
+
= (5.38a)

1
0
2 2
12
4 (2 2 )
n
r
n n
22
2
2
(2 2 )
n
M rh
rvhw −− r rw r vrhw
D
+
++
+ + =− . (5.38b)
5.3.2.4.3 The arc
0
() rr H iv =+ with free edge boundary condition

22
0
22
rt
M
rrrr θ
⎡⎤ ⎛⎞
−+ + =
⎢⎥ ⎜⎟
∂∂ ∂
⎝⎠ ⎣⎦
(5.39a)
2
11 ww w
Dv
∂∂ ∂

22 2
1 1 1 v w w ⎛ ⎞ − ∂ ∂ ∂
0
222 2
11
rt
ww w
DV
rr r r r r rr r θθ θ θ
⎡⎤ ⎛⎞ ∂∂ ∂ ∂
−++ + − =
⎢⎥ ⎜⎟⎜ ⎟
∂∂ ∂ ∂ ∂ ∂∂ ∂
⎝⎠⎝ ⎠ ⎣⎦
(5.39b)
The initial effective shear force is . For the finite mesh points,

0rt
V
22
2
0
22 22
11
2(2 2) 4 (2 2)
n
r
nn n n
M rh
vh w r vrh w r w r vrh w
D
−+
′′+− − ++ =− (5.40a)

23 2 3 2
1 12
33
2
n nn n
Vrh
+−
. (5.40b)
0
32 2 3 2 2 3
112
(2 ) 2 (3 ) (2 ) 4
(2 2 ) ( 2 2 )
n
n
r
nnn
rh v w h v w rh v w r w r hw
rrhrhw rrhrhw rw
D
−
−++
′′ ′′ ′′ −− − − + − − −
++ + +− + − + =−
uations and Boundary Conditions

5.4 Assembly of Governing Eq
The boundary conditions of top and bottom sides plug into the plate governing equation.
If the boundary conditions are different, the and B C matrix will change a little bit.
134
difference scheme substitute into the 1
st
and
nd
mesh line of the plate governing equations. This procedure removes extra
Bottom boundary conditions by central finite
2 unknown
coefficients
00
, ww′′ and
1
w
−
removed. The matrices form shows,

[ ] [ ] [ ] [ ]
[] [ ] [ ]
1,11 1,11 1,2 2 1,1 1
1,2 2 1,3 3 1 ,1 z
Aw B w w C w ′′′′ ′′ ′′ ++ +
(5.41)
1
B
Cw C w w P ++ +Ξ =+ ]
[ ] [ ] [ ] [ ] [ ]
[] [ ] [] [ ]
2,2 2 2,1 1 2,2 2 2,3 3 2,1 1
2,2 2 2,3 3 2,4 4 1 ,2 2 z
Aw B w B w B w C w
Cw C w Cw w P
′′′′ ′′ ′′ ′′ ++ + +
++ + +Ξ = + ] (5.42)
The matrice and s
1,1 1,1 1,2 1,1 1,2 1,3 1
,, , , , , AB B C C C ] 2,2 2,1 2,2 2,3
,, , , A BB B
the Cartesian coordinate systems demonstrate
matrices and
2,1
, C
2,2
C,
2,3
, C
2,4
, C
2
]
in in the Appendix C.1 and C.2; the
1,1 1,1 1,2 1,1 1,2 1,3 1
,, , , , , AB B C C C ] 2,2 2,1
,, AB
2,2 2,3
,, B B
2,1
, C
2,2
C,
2,3
, C
2,4
, C
2
] in
the polar coordinate systems are in the Appendix C.8 and C.9. For the Cartesian
coordinate systems, the constant Ξ is
4
/ ρ . For the polar coordinate systems, the
44
t
hh D
constant is .
The3
rd
~ (n-2)
th
mesh lines by central finite difference scheme of the plate governing
equations shows,
Ξ2/
t
hr h D ρ
[ ] [ ] [ ] [ ]
[] [ ] [ ] [
[] [ ]
,,11 ,,11
,2 2 , 1 1 , , 1 1
,2 2 ,
kk k kk k kk k kk k
kk k kk k kk k kk k
kk k k z k
Aw B w B w B w
Cw C w C w C w
Cw w P
−− + +
−− − − + +
++
′′′′ ′′ ′′ ′′ ++ +
++ + +
++Ξ=
] (5.43)
The matrices in the Cartesian
coordinate system AB B B C
.

,,1 ,,1 ,2 ,1 , ,1 ,2
, , , , , , , , and
kk kk kk kk kk k k kk kk kk z
AB B B C C C C C P
−+ − − + +
s show in Appendix C.3; the matrices
,,1 ,,1 kk kk kk kk −+ ,2 kk −
,1
,
kk
C
− ,,1 ,2
, , and
k k z
CC C P
++
in the polar coordinate systems show in Appendix C.10
,, ,, ,
kk kk
135
central finite difference scheme substitute into the (n-1)
th
Top boundary conditions by
and n
th
mesh lines of the plate governing equations. This procedure deletes extra
unknown coefficients
1 n
w
+
,
2 n
w
+
and
1 n
w
+
′ ′ . The matrices form becomes,
[

] [ ] [ ] [ ] [ ]
[ ] [] [ ] []
1,11 2,1 2 1,11 ,1 3,1 3
2, 1 2 1, 1 1 , 1 1 1
nn n n n n nn n nn n n n n
nn n n n n nn n n z n
Aw B w B w B w C w
Cw C w C w w P
−− − − − − − − − − − − −
−− − − − − − − −
′′′′ ′′ ′′ ′′ ++ + +
++ + +Ξ=+ ]
[
(5.44)
] [ ] [ ] [ ]
[] [ ] [ ]
,,11 , ,22
,1 1 ,
nn n nn n nn n nn n
nn n nn n n z n
Aw B w B w C w
Cw C w w P
−− − −
−−
′′′′ ′′ ′′ ++ +
++ +Ξ=+ ] (5.45)
The matrices
1,1 2,1 1,1 , 1 3, 1 2,1 1,1 ,1
,, , , , , , ,
n n n n nn nn n n n n nn nn
AB B B C C C C
−− − − −− −− − − −−− − ,1
,
zn
P
− 1 n −
] and
,,1 ,
,, ,
nn nn nn
AB B
− ,2
,
nn
C
−

,1 , ,
,, ,
nn nn z n n
CC P
−
] in the Cartesian coordinate systems show in
Appendix C.4 and C.5; the matrices
1,1 2,1 1,1 ,1 3,1 2,1
,, , , , ,
nn n n nn nn n n n n
AB B B C C
−−− −−− − − − − −

1, 1
,
nn
C
−− nn nn nn ,1
,
nn
C
− ,1
,
zn
P
− 1 n −
] and
,,1 ,
,, , A BB
− ,2
,
nn
C
− ,1 , ,
,, ,
nn nn z n n
CC P
−
] in the polar
endix C.11 and C.12.
5.5 State Space Formulations
What the new simple form governing equation as following is combine boundary
conditions with plate governing equations by finite difference scheme along or axis.
coordinate systems demonstrate in App

y r
21
;, , ; ,
nn n
AU BU CU U P A B C U P ω
× ×
′′′′ ′′ ++ −Ξ = ∈ ∈ \\ (5.46)
Where the state variables vectors is defined,
[ ]
1 234 1
(,)(,) (,)(,) (,) (,)
nn
U w xt wxt w xt wxt w xt w xt
−
= " (5.47)
For the Cartesian coo ate systems, gx
T
g θ = . = . For the polar coordinate systems, rdin
136
ed the state variables vectors [ ]
T
UU U U η ′ ′′ ′′′ = After rearrang with Laplace

transformation, the equation (5.46) becomes the first order ordinary differential equation.
4( ) 4( ) 4( ) 1
(, ) ( ) ( , ) (, ); ; ,
nn n
d
dg
gs F s gs q g s F q ηη η
× ×
=+ ∈ ∈ \\ (5.48)
00
()0 0
I

12
000
000
I
F
1
0
I
A
−
⎤
⎢⎥
=
⎢⎥
−
⎣
0
0
q
C I A B ω
−
⎡
⎢⎥
⎢⎥
−Ξ −
⎦
1
0
AP
−
⎡ ⎤
⎢ ⎥
⎢ ⎥
=
⎢ ⎥
⎢ ⎥
⎣ ⎦
ince the top and bottom boundary conditions substitute into the plate governing
quations, those procedures construct a new first order ordinary differential equation.
itions. al with the
other two boundary conditions, left-side and right-side.
R
. (5.49)
S
e
The plate problems have four-side boundary cond Now it is about to de
4( ) 1
() (0, ) () ( , ) ( ); ,
n
Ms sNsms Rs ηη η
×
+= (5 ∈ \ .50)

4( ) 4( )
;
00 0
0 0
,0 ;
sub sub b sub
nn
nn
subi
MM M M
M
×
×
⎡⎤
⎢⎥
⎢⎥
⎥
⎦
∈ \ (5.51a)

42 44
( 3,4; 1,2,3,4)
00 0
00 0
() ; ;
,0 ;
sub sub sub
sub sub sub
nn
subij i j
Ns N
N N
NN N N
N
×
×
==
⎡⎤
⎢⎥
⎢⎥
=∈
⎥
⎦
∈ \ (5.51b)
atrice
11 12 13 14 su
21 22 23 24
() ;
sub sub sub sub
MM M M
Ms M
⎢⎥
=∈ \ 0
0
⎢
⎣
0
( 1,2; 1,2,3,4) j i j ==
4( )nn 4( )
31 32 33 34
41 sub⎣ 43
0
0
sub
N⎢
⎢⎥
N
\ s M The m and in the Cartesian coordinate systems are in Appendix C.6 and C.7;
the matrices
N
M and in the polar coordinate systems are in Appendix C.13 and C.14.

N
137
DTFM
and
quilibrium equations. The finite difference method separates the plate governing
5.6 Solution by
The plate governing equations grained from constitutive, kinematics, resultant
e
equations along y or direction, then top and bottom boundary conditions plug into the
res ma P
M, m . By
rearra se m th the d R
matrices. Applying distributed transfer function method on those two equations, the
closed form result shows in (2.39). If the system is static, the time parameter is equal to
zero.

5.7 Static Numerical Examples
Several numerical examples of finite difference distributed transfer function method on
plate problems present next. The example 5.1 is a rectangular plate with boundary
in the Cartesian coordinate systems. The example 5.2 is a rectangular
late with bou The example
.3 is an annular sector plate with boundary conditions in polar coordinate
ti 32
r
r
semi-finite difference plate governing equations. Those procedu ke A, B, C, and
matrices. The other two side boundary conditions build N and R atrices
nging the state variables tho atrices become F, q wi same M, N, an
CCCC conditions
p ndary conditions CSCF in the Cartesian coordinate systems.
CCCC 5
systems. The example 5.4 is an annular sector plate with boundary conditions CSCF in
the polar coordinate systems. The choice of material is steel. The density of steel is
7850
3
/ Kg m . The modulus of elasticity is 200 GPa . The Possion’s ra o is 0.
(Hibberler, 1994). The length and height a m . The external force is1 N .

e1
138
xample 5.1
This example shows the investigation of FDDTFM method on a in the Cartesian
coordinates system. The notation region means that edge is clamped ,
edge
E
CCCC
CCCC 0 x = ( ) C
0 y = is clamped ( ) C , edge x L = is clamped ( ) C , and edge y H = is clamped ( ) C .
The investigation of the on A-B line shows that F esh lines have
very close appr hows th 5, 35, and 85
mesh lines convergent tendency.

CCCC DDTFM 85 m
oach with FEM 400 elements. The Figure 5.4 s e

Figure 5.3 a square region plate with CCCC boundary conditions
0 0.1 0.2 0.3 0.4 0.5 0.6 0.7 0.8 0.9 1
-2
0
2
4
6
8
10
12
x 10
-11
139
x
ω Deflection

C C C C (Plate-Cartesian)
ω DDTFM 5 F
ω DDTFM 35 F
ω FDDTFM 85
ω FEM 20X20

Figure 5.4 distribution of deflection on A-B for the rectangular region

Table 5.1 deflection of the rectangular plate at

Method Mesh
CCCC
CCCC (0.5,0.5)
11
10 ω ×
5 9.83 FDDTFM
35 7.22
85 6.97
20 20 × FEM 6.83
Difference 2%
85
400
1 100
FDDTFM
FEM
−×

At middle point on A-B line denotes . Table 5.1 shows the difference
between FDDTFM (85) and FEM (400) is very small. The error of deflection is 2%.

( 0.5, 0.5) xy == ( ) C
140
Example 5.2
This example shows the investigation of FDDTFM method on a in the Cartesian
coordinates system. The notation region means that edge is clamped ,
is simple supported , edge
CSCF
CSCF 0 x = ( ) C
edge 0 y = ( ) S x L = is clamped , and edge ( ) C y H = is free
end .

( ) F

Figure 5.5 a square region plate with boundary conditions

The investigation of the on A-B line shows that FDDTFM 15 mesh lines have
very close approach with FEM 2500 elements. The Figure 5.4 shows the 5, 15, and 15
mesh lines convergent tendency. The FDDTFM convergence is guaranteed.
At middle point on A-B line denotes . Table 5.2 shows the difference
between 15%.

CSCF
CSCF
( 0.5, 0.5) xy == ( ) C
FDDTFM (15) and FEM (2500) is very small. The error of deflection is 1.
0 0.1 0.2 0.3 0.4 0.5 0.6 0.7 0.8 0.9 1
-2
0
2
16
x 10
-11
141
4
6
8
10
12
14
ω Def
x

lection
CSCF sian)

(Plate-Carte
ω FDDTFM 5
ω FDDTFM 11
ω FDDTFM 15
ω FEM 50X50

Figure 5.6 distribution of deflection on A-B for the rectangular region

Method Mesh
CSCF

Table 5.2 deflection of the CSCF rectangular plate at (0.5,0.5)
11
10 ω ×
5 15.15
11 12.83
FDDTFM
15 12.23
FEM 50 50 × 12.09
Difference
15
2500
1 100
FDDTFM
−×
1.15%
FEM

Example 5.3
This example shows the investigation of FDDTFM method on a CCCC in the polar
coordinates system. The notation CCCC region means that edge /2 θ π = is
clamped , edge is clamped () C
i
rr = () C , edge 0 θ = is clamped() C , and edge
o
rr = is
142
igure 5.8 shows the
, 35, a 7 mesh lines convergent tendency.
t . Table 5.1 shows the difference

clamped ( ) C . The investigation of the CCCC on A-B line shows that FDDTFM 147
mesh lines have very close approach with FEM 1800 elements. The F
5 nd 14
( 0.5, 0.5) xy == ( ) C At middle poin on A-B line denotes
between FDDTFM (147) and FEM (1800) is very small. The error of deflection is 3.05%.

Figure 5 ul l da io
ble 5 on ar te
.7 an ann ar sector p ate with CCCC boun ry condit ns

Ta .3 deflecti of the CCCC annul sector pla at (1.5, / 3) π

Method Mesh
10
10 ω ×
5 4.00
35 1.59
FDDTFM
147 1.35

30 60 × FEM 1.31
Difference
147
FDDTFM
−×
3.05%
1800
1 100
FEM

143
0 pi/6 pi/3 pi/2
0
0.5
1
1.5
2
2.5
3
3.5
4
4.5
x 10
-10
C C r)

C C(Plate-Pola
x

ω FDDTFM 5
ω FDDTFM 35
ω FDDTFM 147
ω FEM 30X60
ω Deflection

Figure 5.8 distribution of deflection on A-B for the annular sector plate

Example 5.4
CCCC

Fig 5.9 r a nd ure an annula sector pl te with CSCF bou ary conditions
0 pi/6 pi/3 pi/2
0
1
2
7
x 10
-9
144
3
4
5
6
ω D on
C S C F(Plate-Polar)

x

eflecti
ω FDDTFM 15
ω FDDTFM 45
ω FDDTFM 77
ω FEM 30X60

Figure 5.10 distribution of deflection on A-B for the annular sector plate

annular sector plate at
CSCF
CSCF (1.5, / 3) π Table 5.4 deflection of the

Method Mesh
9
0 ω × 1
5 4.85
45 2.28
FDDTFM
77 1.60
FEM 30 0 6 × 1.59
Difference
77
1 100
FDDTFM
FEM
−×
1800
0.6%

This example shows the investigation of FDDTFM method on a in the polar
coordinates system. The notation region means that edge
CSCF
CSCF /2 θ π = is clamped ,
edge is simple supported , edge
( ) C
i
rr = () S 0 θ = is clamped , and edge is free
he investigation of the on A-B line shows that FDDTFM 77 mesh lines
() C
o
end ( ) F . T
rr =
CSCF
145
ave very close approach with FEM 1800 elements. The Figure 5.10 shows the 15, 45,
nd 77 mesh lin DTFM ranteed.
At specific point 0.5 x== A-B denotes . Table 5.4 shows the
difference betw n FD 7 M i m e eflection
is 0.6%.

5.8 Dynamic um
er app f f s tr n ethod on
ple 5.5 is a rectangular plate with boundary
conditions in the Cartesian coordinate systems. The exampl
with boundary conditions in the Cartesian coordinate systems. The height is 1
and the width is 1.5. The example 5.7 is an annular sector plate with boundary
conditions in the Polar coordinate systems. The example 5.
with boundary conditions in the Polar coordinate systems. The outer radius is 2
and the inner radius is 1. The following results show the non-dimensional number.

Example 5.5
This example shows the investigation of FDDTFM method on a in the Cartesian
oordinates system. The notation region means that edge is simple
pported , edge is simple supported , edge
h
a es convergent tendency. The FD convergence is gua
( 0.5,y ) on line ( ) C
ee DTFM ( 7) and FE (1800) s very s all. The rror of d
N erical examples
Several num ical roaches o finite dif erence di tributed ansfer fu ction m
the plate show next. The exam SSSS
e 5.6 is a rectangular plate
CCCC
SSSS
6 is an annular sector plate
CCCC
SSSS
SSSS 0 x = c
() S 0 y = () S x L = su is simple supported ,
nd edge is simple supported . The investigation of the on natural
equency of free vibration shows that FDDTFM 32 mesh lines have very close approach
() S
yH = () S SSSS a
fr
146
ith FEM 1024 elements. The Table 5.5 shows the 8, 16, and 32 mesh lines convergent
ndency. The FDDTFM convergence is guaranteed. Those results also compare with
EM.
Table 5.5 natural frequency of a rectangular plate

FDDTFM FEM Exact
w
te
F
SSSS

Mode no.
8 16 32 88 × 32 32 ×
1 13.9631 14.1805 14.2439 14.0801 14.2449 14.256
2 27.2350 27.4063 27.2744 26.7395 27.3710 27.416
3 41.8894 43.2773 43.7738 43.1889 43.8199 43.865
4 49.0783 49.3611 49.3548 47.9405 49.2484 49.348
5 54.8848 56.5031 56.8203 54.3757 56.8454 57.024
6 77.0046 78.4579 78.7947 73.2926 78.5569 78.957
7 79.2166 80.0450 80.0461 77.8959 79.8780 80.053
8 83.3641 90.7068 92.6783 91.8421 93.1123 93.213
9 96.6359 103.8515105.7299100.3558105.9705 106.37
10 107.6959 108.6128 109.5515 100.6971 108.9567 109.66

Example 5.6

Table 5.6 natural frequency of a rectangular plate CCCC
FDDTFM FEM Lessia Mode no.
8 16 32 88 × 32 32 ×
1 25.5760 26.7252 26.7857 26.4579 26.9678 27.010
2 40.3226 41.3974 41.6475 40.3057 41.5849 41.716
3 58.7558 64.1543 65.5846 63.7743 66.0301 66.143
4 65.2074 66.1688 66.5323 64.8368 66.3047 66.552
5 72.5806 77.5922 79.3203 75.0001 79.4500 79.850
6 96.0829 100.3726 100.7684 93.7213 100.4809 100.85
7 99.7696 101.0136 102.5801 97.3619 102.3883
8 101.6129 118.4508125.2880122.1938125.1250
9 114.9770 126.5284135.4803123.2770134.8759
10 129.2627 131.6569136.8142130.1474137.9693

This example shows the investigation of FDDTFM method on a CCCC in the Cartesian
coordinates system. The notation CCCC region means that edge 0 x = is clamped ( ) C ,
edge 0 y = is clamped ( ) C , edge x L = is clamped ( ) C , and edge y H = is clamped ( ) C .
The investigation of the CCCC on natural frequency of free vibration shows that
FDDTFM 32 mesh lines have very close approach with FEM 1024 elements. The Table
5.6 shows the 8, 16, and 32 mesh lines convergent tendency. The FDDTFM convergence
is guaranteed. Those results also compare with FEM.

Example 5.7
Table 5.7 natural frequency of aCSCF annular sector plate

FDDTFM FEM % Δ Mode no.
18 30 50 30 60 × 35 70 ×
1 5.4147 6.3364 7.4885 7.5122 7.5207 0.43%
2 12.788014.170515.553014.768 14.774 4.22%
3 23.6175 26.382526.757 26.756 1.42% 25.0000
4 37.9032 40.898638.776 38.848 1.13% 39.2857
5 52.1889 42.972442.722 42.713 0.59% 40.6682
6 54.0323 58.640649.422 49.713 6.37% 52.8802
7 55.8756 61.866462.387 62.367 0.81% 57.2581
8 59.5622 74.769666.953 67.016 10.3% 60.7143
9 71.3134 80.069185.733 85.692 7.02% 72.9263
10 72.4654 91.129090.621 90.667 0.51% 78.6866

This example shows the investigation of FDDTFM method on a in the polar
coordinates system. The notation region means that edge is
supported , edge is simple supported , edge
CSCF
CSCF
0
90 θ = simple
() S
i
rr = () S
0
0 θ = is simple supported ,
is simple supported
on natural frequency of free vibration shows that
DDTFM 50 mesh lines have very close approach with FEM 2450 elements. The Table
.7 shows the 18, 30, and 50 mesh lines convergent tendency. The FDDTFM
() S
o
rr = () S . and edge
The investigation of the CSCF
F
5
147
148
onvergence is guaranteed. Those results also compare with FEM. The definition of
is the closest difference between FDDTFM and FEM on the Table. The result shows
is less than 3.28%.

Example 5.8
his example shows the investigation of FDDTFM method on a in the polar
oordinates system. The notation region means that edge is clamped ,
dge is clamped , edge
c
% Δ
the average % Δ
CCCC T
CCCC
0
90 θ = () C c

i
rr = () C
0
0 θ = e is clamped , and edge is clamped .
of the on natural frequency of free vibration shows that
DDTFM 50 mesh lines have very close approach with FEM 1800 elements. The Table
onvergence is guaranteed. Those results also compare with FEM. The definition of
ble. The result shows
() C
o
rr = () C
CCCC The investigation
F
5.8 shows the 18, 30, and 50 mesh lines convergent tendency. The FDDTFM
c
% Δ is the closest difference between FDDTFM and FEM on the Ta
the average % Δ is less than 0.732%.

Table 5.8 natural frequency of aCCCC annular sector plate
FDDTFM FEM % Δ Mode no.
18 30 50 816 × 30 60 ×
1 13.940115.080615.783415.63415.724 0.38%
2 17.396318.467719.009218.79218.975 0.18%
3 23.617524.596825.230424.53124.843 0.16%
4 32.834133.790334.216632.76033.225 1.20%
5 36.751239.758141.589941.33541.792 0.49%
6 39.977043.145245.046143.13643.784 1.48%
7 45.046145.725846.428644.59345.283 0.53%
8 46.198249.112951.036950.29151.345 0.60%
9 54.953957.661359.486055.30256.172 2.21%
10 59.792660.564560.714358.67160.180 0.09%

149
.9 Discussion
o
h a
5
The absolute value of difference between FDDTFM and FEM divided by FEM defines as
an err r (difference). It is easy to check boundary conditions on the edge from the
original definition such as (2.21; 2.22; 2.23) in the Cartesian coordinate systems and
(2.30; 2.31; 2.32; 2.33) in the polar coordinate systems. In FDDTFM, the solution obtains
four parameters, suc s , , , UU U U ′ ′′ ′′′ . The deflection, slope, bending moment and shear
force can easily obtain from those four parameters. The fewer mesh lines got the same
accurate solution on each case. (Example 5.1~5.8) The computation of natural frequency
by FDDTFM is fewer mesh lines than by FEM mesh elements. (Table 5.5~5.8)

150
ry conditions and external factors.
region case, there are at least 6561 possibilities. Each edge has
ctor region case also has at least 6561 possibilities. In the 2-D heat conduction
ach edge has 4 possible boundary conditions, but adiabatic boundary condition cannot
lux boundary conditions. In the 2-D heat conduction
gions. In the rectangular plate problems, there are at least 81 possible cases. Each edge
1 possible cases. The proposed method doesn’t need solve different boundary conditions
d-form, semi-analytic solution on the two-
TFM doesn’t need to look for a complicated series function.
ompute integral by part procedure. This dissertation shows the FDDTFM is much faster
Chapter 6

Conclusions

The FDDTFM shows flexibility with arbitrary bounda
Any kind of boundary conditions in appendices A, B, C makes more possible cases. In
the 2-D elastic rectangular
3 possible boundary conditions and 3 possible loading options. The 2-D elastic annular
se
problems on this dissertation, there are at least 192 possibilities on the rectangular regions.
E
put on the opposite side of heat f
problems on this dissertation, there are at least 192 possibilities on the annular sector
re
has 3 possible boundary conditions. In the annular sector plate problems, there are at least
8
case by case. The FDDTFM provides a close
dimensional elastic continua problems. Especially in multi-body region with mixed
boundary conditions, the FDD
The FDDTFM doesn’t have to evaluate eigenvalues or natural frequencies directly;
c
151
convergence than the FEM in three representations such as, 2-dimensional elasticity, 2-
dimensional heat conduction and plate problems.

152
ibliography
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& Uddin, M. W. (2000). A finite-
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nd
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rd
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B

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161
APPENDICES
Appendix A section contains 10 appendices for two-dimensional elasticity problems. The first fiv e Cartesian
coordinate systems. The last five appendices are for the polar coordinate systems. Each appendix covers displacement, traction, and
mixed boundary conditions.

Appendix A.1: The bottom mesh line for two-dimensional elasticity problems in the Cartesian coordinate systems

Bottom mesh line
e appendices are for th
Boundary Conditions Displacement Traction Mixed (example)
1,1
A
2
10
1
0
2
h
v
⎡⎤
⎢⎥
−
⎢⎥
⎣⎦

2
10
1
0
2
h
v
⎡ ⎤
⎢ ⎥
−
⎢ ⎥
⎣ ⎦

2
10
1
0
2
h
v
⎡ ⎤
⎢ ⎥
−
⎢ ⎥
⎣ ⎦

1,2
B
01 (1 )
10 4
vh ⎡ ⎤ +
⎢ ⎥
⎣ ⎦
01 (1 )
10 4
vh ⎡ ⎤ +
⎢ ⎥
⎣ ⎦

01 (1 )
10 4
vh ⎡ ⎤ +
⎢ ⎥
⎣ ⎦

1,1
C
(1 ) 0
02
v −−⎡⎤
⎢⎥
−
⎣⎦

22 2
22 22 22
22 2
⎡
22 22 22
(1 ) 2 (1 ) 1 3
24 4
13 2 (1 )
44
vL vh v vh v hL
Lvh L vh Lvh
vh L vh v h
Lvh Lvh Lvh
⎤ −− + −
−
+
L
⎢ ⎥
−− −
⎢ ⎥
⎢ ⎥ −−++
−−
⎢ ⎥
−− − ⎦

⎣
53
0
4
2
v
L
h
−+ ⎡ ⎤
⎢ ⎥
⎢ ⎥
−
⎢ ⎥
−
⎢ ⎥
⎣ ⎦

1,2

C
(1 )
0
v −⎡⎤

2
01
⎢⎥
⎢⎥
⎣⎦
(1 )
0
v −
2
01
⎡ ⎤
⎢ ⎥
⎢ ⎥
⎣ ⎦
(1 )
0
v −
2
01
⎡ ⎤
⎢ ⎥
⎢ ⎥
⎦

⎣
1
P
0
0
1
1
2
xb
v
u
−⎡
−+
⎢⎥
1 yb
A
vB
⎢⎥
−+
⎢⎥
⎣⎦

⎤
00
0 0
222
1 2 2
)
)
bb
xy
vhL
ppA
vh E
2
(1 3 ) 2(1 )
4(
v
Lv
2
222 2
1 22 22
(1 (1 3 ) (1
()
) 4 (
(1 3) 2( ) (1 (1 ) (1 )(1 )
()
4( ) 4 ( ) 4
b b
x y
vhL h v
h E L
hL v vhL v vh
) 2(1 )
4
v v
E
1 vv 3)
p pB
Lvh E Lvh vE vE
⎡ ⎤ −
+
− + −
+
+ +
+
⎢ ⎥
−−
⎢ ⎥
⎢ ⎥
− + −
−+− + +
−+ −
00
00
2
1
1
) (3 )
24
2(1 )
xb xb
xb xb
vh v (1
p uA
E
vL
Lp u B
Eh
⎡ ⎤ +−
−− +
⎢ ⎥
⎢ ⎥
+
⎢ ⎥
−−+
⎢ ⎥ ⎢ ⎥
−−
⎣ ⎦
⎣ ⎦
11
22 2 22 2
11
1122
(1 ) (1 )
(); ( )
xy
vh u v h v
A fB f
E tE t
ρρ
−∂ −∂
=− − + =− − +
∂ ∂

162
Appendix A.2: The esh lines for two-dimensional elasticity problems in the Cartesian coordinate systems

2
nd
~(n-1)
th
me
m
sh lines
Boundary Conditions Any case
, kk
A
2
10
(1 )
0
2
h
v
⎡ ⎤
⎢ ⎥
−
⎢ ⎥
⎣ ⎦

,1 kk
B
−

01 1
10 4
v
h
− ⎡⎤ +
⎢⎥
−
⎣⎦

,1 kk
B
+

01 1
10 4
v
h
⎡⎤ +
⎢⎥
⎣⎦

,1 kk
C
−

1
0
2
01
v − ⎡ ⎤
⎢ ⎥
⎢ ⎥
⎣ ⎦

, kk
C
(1 ) 0
02
v −− ⎡ ⎤
⎢ ⎥
−
⎣ ⎦

,1 kk
C
+

1
0
2
01
v − ⎡ ⎤
⎢ ⎥
⎢ ⎥
⎣ ⎦

k
P
2 22
2
2 22
2
(1 )
()
(1 )
()
k
xk
k
yk
u vh
f
Et
v vh
f
Et
ρ
ρ
⎡ ⎤ ∂ −
−− +
⎢ ⎥
∂
⎢ ⎥
∂ ⎢ ⎥ −
−− +
⎢ ⎥
∂ ⎣ ⎦

163
Appendix A.3: The top mesh line for two-dimensional elasticity problems in the Cartesian coordinate systems

Top mesh line
Boundary
Conditions
Displacement Traction Mixed (example)
, nn
A
2
10
1
0
2
h
v
⎡⎤
⎢⎥
−
⎢⎥
⎣⎦

2
10
1
0
2
h
v
⎡ ⎤
⎢ ⎥
−
⎢ ⎥
⎦

⎣
2
10
1
0
2
h
v
⎡ ⎤
⎢ ⎥
−
⎢ ⎥
⎣ ⎦

,1 nn
B
−

10 (1 )
01 4
vh −⎡⎤ +
⎢⎥
−
⎣⎦
10 (1 )
01 4
vh − ⎡ ⎤
10 (1 )
01 4
vh − ⎡ ⎤ +
⎢ ⎥
−
⎣ ⎦

+
⎢ ⎥
⎦

−
⎣
,1 n − n
C
(1 )
0
2
01
⎢⎥
⎢⎥
⎣⎦
v −⎡⎤ (1 )
0
2
01
v − ⎡ ⎤
⎢ ⎥
⎢ ⎥
⎣ ⎦
(1 )
0
2
01
v − ⎡ ⎤
⎢ ⎥
⎢ ⎥
⎣ ⎦
, nn
C
(1 ) 0
02
v −−⎡⎤
⎢⎥
−
⎣

⎡
22 2
22
2 (1 )
4
L vh v vh
Lvh
22
2
13
Lvh
vh
2 2
2
22 22 22
(1 ) 1 3
4
)
44
v v hL
vh
L v h
22
2 (1
L
L vh
⎤ −− −
−
+ +
−
⎢ ⎥
− − −
⎢
⎦
⎥
⎢
L vh L vh L vh
⎥ −− +
−
+
⎢ ⎥
−− − ⎦

⎣
35
0
4
2
v
L
h
−+ ⎡ ⎤
⎢ ⎥
⎢ ⎥
⎢ ⎥
−
⎢ ⎥
⎣ ⎦

n
P
0
0
1
2
xb n
yb n
v
uA
vB
−⎡⎤
−+
⎢⎥
⎢⎥
−+
⎢⎥
⎣⎦

11
1 1
222
22 22
22
⎡
2
(1
2
22 22
(1 3 ) 1 ) (1 ) 2(1 ) (1 3 ) (1 )
()
4( ) 4 4( )
(1 3 ) 2 ) (1 3 ) (1 ) (1 ) )
()
4( ) 4( ) 4
nn
n n
xt yt n
xt yt n
vvhL v vh v vhL 2(
( 1
p pA
Lvh E E vh E
vvhL v vhL vh
L
v
p pB
Lvh E Lvh vE vE
++
+ +
⎤ −+ + + − −
−− + +
⎢ ⎥
−−
⎡
⎢ ⎥
⎢ ⎥
− − −
−+ − +
−+ +
⎢ ⎥
−−
⎣ ⎦

11
11
2
(1 ) 1 3
24
2(1 )
nn
nn
xt xt n
xt xt n
vh v
p uA
E
vL L
pu B
Eh
++
++
⎤ +−
−− +
⎢ ⎥
⎢ ⎥
+
⎢ ⎥
−+ +
⎢ ⎥
⎣ ⎦

22 2 22 2
22
(1 ) vh (1 )
(); ( )
nn
nn
nxn y
u vh v
A fB f
E tE t
ρρ
−∂ −∂
=− − + =− − +
∂

∂

164
Appendix A.4: The left side mesh points for two-dimensional elasticity problems in the Cartesian coordinate systems

0 ( ) x i = line
Boundary Conditions Displacement Traction Mixed (example)
1,1
M
22
22 22
2
22 22
vhL vL
L vh L vh
LhL
L vh L vh
⎡ ⎤
⎢ ⎥
−−
⎢ ⎥
⎢ ⎥
⎢ ⎥
−− ⎣ ⎦
1,1
M
0
01
vL
h
⎡ ⎤
⎢ ⎥
⎢ ⎥
⎣ ⎦

,1 kk
M
−
0
10
v − ⎡ ⎤
⎢ ⎥
−
⎣ ⎦

,1 kk
M
−

0
00
v − ⎡ ⎤
⎢ ⎥
⎣ ⎦

,1 kk
M
+
0
10
v ⎡ ⎤
⎢ ⎥
⎣ ⎦

,1 kk
M
+

0
00
v ⎡ ⎤
⎢ ⎥
⎣ ⎦

1 sub
M I
, nn
M
22
vhL vL
22 22
22
2
22
L vh L vh
hL L
L vh L vh
⎡ ⎤ −
⎢ ⎥
−−
⎢ ⎥
⎢ ⎥ −
⎢ ⎥
− − ⎣ ⎦
, nn
M 0
vL
01
h
⎡ ⎤
⎢ ⎥
⎢ ⎥
⎦

⎣
2 sub
M
0
, kk
M
2 0
h 02
h ⎡ ⎤
⎢ ⎥
⎦

, kk
M
⎣
20
00
h ⎡ ⎤
⎢ ⎥
⎣ ⎦

3 sub
M
0 0 0
4 sub
M
0 0 0

165
Appendix A.5: The right side mesh points for two- ensional elasticity problems in the Cartesian coordinate systems

dim
() x L iii = line
Boundary Conditions Displacement Traction Mixed (example)
0 0 0
1 sub
N

2 sub
N
0 0 0
1,1
N
22
22 22
2
22 22
vhL vL
L vh L vh
LhL
L vh L vh
⎡ ⎤
⎢ ⎥
−−
⎢ ⎥
⎢ ⎥
⎢ ⎥
−− ⎣ ⎦
1,1
N
0
01
vL
h
⎡ ⎤
⎢ ⎥
⎢ ⎥
⎣ ⎦

,1 kk
N
−
0
10
v − ⎡ ⎤
⎢ ⎥
−
⎣ ⎦

,1 kk
N
−

0
00
v − ⎡ ⎤
⎢ ⎥
⎣ ⎦

,1 kk
N
+
0
10
v ⎡ ⎤
⎢ ⎥
⎣ ⎦

,1 kk
N
+

0
00
v ⎡ ⎤
⎢ ⎥
⎣ ⎦

3 sub
N I
, nn
N
22
22 22
2
22 22
vhL vL
L vh L vh
LhL
L vh L vh
⎡ ⎤ −
⎢ ⎥
−−
⎢ ⎥
⎢ ⎥ −
⎢ ⎥
−− ⎣ ⎦
, nn
N 0
01
vL
h
⎡ ⎤
⎢ ⎥
⎢ ⎥
⎣ ⎦

4 sub
N
0
, kk
N
20
02
h
h
⎡ ⎤
⎢ ⎥
⎣ ⎦

, kk
N
20
00
h ⎡ ⎤
⎢ ⎥
⎣ ⎦

166
Appendix A.6: The bottom mesh line for two-dimensional elasticity problems in the polar coordinate systems

Bottom mesh line
Boundary Conditions Displacement Traction Mixed (example)
1,1
A
2
1
0
2
01
v
h
− ⎡ ⎤
⎢ ⎥
⎢ ⎥
⎣ ⎦

2
h
1
0
2
01
v − ⎡ ⎤
⎢ ⎥
⎢ ⎥
⎦

2
1
0
2
01
v
h
− ⎡ ⎤
⎢ ⎥
⎢ ⎥
⎣ ⎦

⎣
1,1
B
2
01 (3 )
10 2
vh − ⎡ ⎤ −
⎢ ⎥
⎣ ⎦

2
01 (3 )
10 2
vh − ⎡ ⎤ −
⎢ ⎥
⎣ ⎦

2
01 (3 )
10 2
vh − ⎡ ⎤ −
⎢ ⎥
⎣ ⎦

1,2
B
01 (1 )
10 4
vrh ⎡ ⎤ +
⎢ ⎥
⎣ ⎦
01 (1 )
10 4
vrh ⎡ ⎤ +
⎢ ⎥
⎣ ⎦
01 (1 )
10 4
vrh ⎡ ⎤ +
⎢ ⎥
⎣ ⎦

1,1
C
22
10
(2 )
1
0
2
rh
v
⎡ ⎤
⎢ ⎥
−+
−
⎢ ⎥
⎦

11 12
21 22
ΩΩ ⎡ ⎤
⎢ ⎥
ΩΩ
⎣ ⎦

2
22
2
(1 )
10 0
4
(2 )
1
0 (1 )(2 )
0 2
4
vr
v
rh
v
vr hr θ
⎡ ⎤
⎡⎤
+
⎢ ⎥
⎢⎥
⎢ ⎥ +
−
⎢⎥
−+
⎢ ⎣
vh
⎥
⎣⎦
−−
−
⎢ ⎥
⎣ ⎦
1,2
C
2
10
()
1
2 0
2
rh
r
v
⎡ ⎤
⎢ ⎥
+
−
⎢ ⎥
⎣ ⎦

2
10
()
1
2 0
2
rh
r
v
⎡ ⎤
⎢ ⎥
+
−
⎢ ⎥
⎣ ⎦

2
10
()
1
2 0
2
rh
r
v
⎡ ⎤
⎢ ⎥
+
−
⎢ ⎥
⎦

⎣
1
1
1
1
r
PA
PB
θ
+ ⎡ ⎤
⎢ ⎥
+
⎢ ⎥
⎣ ⎦

1
1
1
1
mr
m
P A
P B
θ
+ ⎡ ⎤
⎢ ⎥
+
⎢ ⎥
⎣ ⎦

1
P
0
0
2
1
2
1
()
2
()
22
rb
b
rh
ru A
vrh
rv B
θ
⎡ ⎤
−+
1
⎢ ⎥
⎢ ⎥
−
⎢ ⎥
−+
⎢ ⎥
⎦

⎣

1
22 2 2
1
1
2
(1 )
()
r
vrh u
Af
Et
ρ
−∂
=− − +
∂
1
22 2 2
1
1
2
(1 )
()
vrh v

167
B f
Et
θ
ρ
−∂
=− − +
∂
22 2 2
22
11
22
4[ ( )( ) ] vh r vh r h θ −− +
(1 ) 2(2 )( )
(2 )
vr h r r h r
rh
θ +− − +
Ω= − +
h
2
12
22
[(1 )( ) h 2(2 ) ]
4[ ( )( ) ]
vr v r hvrh
vh r vh r h θ
+− − −
Ω=
−− +
θ

2
2 )] r h r
21
22
1 )(
4[ ( )( ) ]
v h
vh r vh r h
[( )( ) (1 vr h θ
θ
++ − −
Ω=
−− +

−
222
22
22
22
[(1 ) (1 )(2 )( ) ] (1 )
(2 )
4[ ( )( ) ] 2
vvh v r h r vh r v
rh
vh r vh r h
θ
θ
+−− − − −
Ω= − +
−− +

1 0 0
222 22
22 22
[(1 ) 2(2 )( ) ](1 ) [(1 )( ) 2(2 ) ](1 )
{}{ }
4[ ( )( ) ] 2[ ( )( ) ]
r rb b
vh r h r h v r h v r vh r h v vr h
Pp p
vh r vh r h E vh r vh r h E
θ
θθ
θθ
+− − + − + − − − +
=+
−− + −− +
1 0 0
22 2 2 2
22 22
[(1 )( ) (1 )(2 )](1 ) [(1 ) (1 )(2 )( ) ](1 )
{}{ }
4[ ( )( ) ] 2[ ( ) h ( ) ]
rb b
vr h v r h v rh vvh v r h r vh vrh
Pp p
vh r vh r h E vh r v r h E
θ
θθ
θθ
++ − − − − + − − − − +
=+
−− + − − +

θ
1 00
22
(1 )( ) (2 ) (1 )(1 )
[]
42 4
mr rb rb
vr vhr r hr v v rh
Pu p
vvE
+− − + −
=− +
10 0
22
r θθ (1 )(2 )( ) (1 )(2 ) )
44
mrb rb
vr hr v r vr h v
Pu p
vh vE
θ
−− − − − −
=− −
(1 h

168
Appendix A.7: The mesh lines for two-dimensional elasticity problems in the polar coordinate systems
2
nd
~(n-1)
th
mesh lines

Boundary Conditions Any case
, kk
A
2
(1 )
0
2
01
v
h
− ⎡ ⎤
⎢ ⎥
⎢ ⎥
⎣ ⎦

,1 kk
B
−

2
01 1
10 4
v
h
− ⎡⎤ +
⎢⎥
−
⎣⎦

, kk
B
2
01 3
10 4
v
h
− ⎡⎤ −
⎢⎥
⎣⎦

,1 kk
B
+

2
01 1
10 4
v
h
⎡⎤ +
⎢⎥
⎣⎦

,1 kk
C
−

2
10
()
1
2 0
2
rh
r
v
⎡⎤
⎢⎥
−
−
⎢⎥
⎣⎦

, kk
C
22
10
(2 )
1
0
2
rh
v
⎡⎤
⎢⎥
−+
−
⎢⎥
⎣⎦

,1 kk
C
+

2
10
()
1
2 0
2
rh
r
v
⎡⎤
⎢⎥
+
−
⎢⎥
⎣⎦

k
P
22 2 2
2
22 2 2
2
(1 )
()
(1 )
()
k
rk
k
k
vrh u
f
Et
vrh v
f
Et
θ
ρ
ρ
⎡ ⎤ −∂
−− +
⎢ ⎥
∂
⎢ ⎥
⎢ ⎥ −∂
−− +
⎢ ⎥
∂ ⎣ ⎦

169
mesh line
Appendix A.8: The top mesh line for two-dimensional elasticity problems in the polar coordinate systems

Top
Boundary Displacement Traction Mixed (example)
Conditions
, nn
A
2
1
0
2
01
v
h
− ⎡ ⎤
⎢ ⎥
⎢ ⎥
⎣ ⎦

2
1
0
2
01
v
h
− ⎡ ⎤
⎢ ⎥
⎢ ⎥
⎣ ⎦

2
1
0
2
01
v
h
− ⎡ ⎤
⎢ ⎥
⎢ ⎥
⎣ ⎦

,1 nn
B
−

2
01 1
10 4
v
h
− ⎡ ⎤ +
⎢ ⎥
−
⎣ ⎦

2
01
10
1
4
v
h
− ⎡ ⎤ +
⎢ ⎥
⎦

−
⎣
2
01 1
10 4
v
h
− ⎡ ⎤ +
⎢ ⎥
−
⎣ ⎦

, nn
B
2
01 (3 )
10 2
vh − ⎡ ⎤ −
⎢ ⎥
⎣ ⎦

2
01 (3 )
10 2
vh − ⎡ ⎤ −
⎢ ⎥
⎦

⎣
2
01 (3 )
10 2
vh − ⎡ ⎤ −
⎢ ⎥
⎣ ⎦

,1 nn
C
−

2
10
()
1
2 0
2
rh
r
v
⎡ ⎤
⎢ ⎥
−
−
⎢ ⎥
⎦

⎣
2
10
()
1
2 0
2
rh
r
v
⎡ ⎤
⎢ ⎥
−
−
⎢ ⎥
⎣ ⎦
2
10
()
1
2 0
2
rh
r
v
⎡ ⎤
⎢ ⎥
−
−
⎢ ⎥
⎣ ⎦

, nn
C
22
10
(2 )
1
0
ΨΨ
2
rh
v
⎡ ⎤
⎢ ⎥
−+
−
⎢ ⎥
⎣ ⎦
21 22
11 12
⎡ ⎤
⎢ ⎥
ΨΨ
⎦

2
22
⎣
2
0 (1 )(2 )
0 2
4
vr hr
vh
θ
(1 )
10 0
4
(2 )
1
vr
v
rh
v
⎡ ⎤ +
⎡⎤
⎢ ⎥
⎢⎥
⎢ ⎥ −+ +
−
⎢⎥
⎢ ⎥ −+
⎣⎦
⎢ ⎥
⎣ ⎦
n
n
mr n
mn
P A
P B
θ
+ ⎡ ⎤
⎢ ⎥
+
⎢ ⎥
⎣ ⎦

n
P
1
2
()
2
n
rt n
ru A
+
1
2
1
()
22
tn
rh
vr
rv B
n
θ
+
h
⎡ ⎤
++
⎢ ⎥
⎢ ⎥
−
⎢ ⎥
++
⎢ ⎥
⎣ ⎦
PB
θ
n
n
rn
n
PA + ⎡ ⎤
⎢ ⎥
+
⎢ ⎥
⎦

⎣

22 2 2
()
n
h u
Af ρ
−∂
− +
2
n
Et
(1 )
n
r
vr
=−
∂

22 2 2
2
(1 )
()
n
n
n
vrh v
B f
Et
θ
ρ
−∂
=− − +
∂

22 2 2
22
11
22
(1 ) 2(2 )( )
(2 )
4[ ( )( ) ]
vr h r h r h r
rh
vh r vh r h
θ
θ
+− + −
Ψ= − +
−+ −

2
12
22
[(1 )( ) 2(2 ) ]
4[ ( )( ) ]
vr vh r hvrh
vh r vh r h
θ
θ
−+ + + +
Ψ=
−+ −

2
21
22
[(1 )( ) (1 )(2 )]
4[ ( )( ) ]
vr h v r h rh
vh r vh r h
θ
θ
−+ − + − +
Ψ=
−

−+
222
22
[(1 ) (1 )(2 )( ) ] (1 )
(2 )
vvh v r h r vh r v
rh
θ +−− + + −
Ψ= − +
22
22
4[ ( )( ) ] 2 vh r vh r h θ −+ −
1 1
222 22
22 22
[ (1 ) 2(2 )( ) ](1
( )( ) ]
r h rh
rvh rh E
) [
{}
rt
rh ( ](1 )
}
4[ ) ]
n n
r t
vh v v v vrh
vh h E
θ
θ
θθ
1 )(
2[
v r
vh
θ ) 2(2 )
{
( )(
h r h
Ppp
rvh r
n + +
−+ + +
−

+ − −
−+ −
+ + − +
−+
=+
1 1
22 2 22
22 22
[(1 )( ) (1 )(2 )](1 ) [ (1 ) (1 )(2 )( ) ](1 )
{}{ }
4[ ( )( ) ] 2[ ( )( ) ]
n n n
rt t
v r h v r h v r h v vh v r h r vh v r h
Pp p
vh rvh rh E vh rvh r h E
θ θ
θθ
θθ
+ +
+− − − + − −+ + − + + +
=+
−+ − −+ −

11
22
(1 )( ) (2 ) (1 )(1 )
[]
42 4
n nn
mr rt rt
vr vhr r hr v v rh
Pu p
vvE
+ +
++ + − −
=− −
11
22
(1 )(2 )( ) (1 )(2 )(1 )
44
nn n
mrt rt
vr hr vhr vr h v r
Pu p
vh vE
θ
θθ
+ +
−+ + −+ −
=−

170

171
Appendix A.9: The right side mesh points for two-dimensional ela problems in the polar coordinate systems

sticity
0 ( ) iii θ = line
Boundary Conditions Displacement Traction ample) Mixed (ex
1,1
M
11 12
21 22
α α
α α
⎡ ⎤
⎢ ⎥
⎦

⎣
1,1
M
10
00
⎡ ⎤
⎢ ⎥
⎣ ⎦

,1 kk
M
−
0
0
r
vr
− ⎡ ⎤
⎢ ⎥
−
⎣ ⎦

,1 kk
M
−

00
0 vr
⎡ ⎤
⎢ ⎥
−
⎦

⎣
, kk
M
02
20
h
h
− ⎡ ⎤
⎢ ⎥
⎦

M
⎣
, k k

10
20 h
⎡ ⎤
⎢ ⎥
⎣ ⎦

,1 kk
M
+
0
0
r
vr
⎡ ⎤
⎢ ⎥
⎦

⎣
,1 kk
M
+

00
0 vr
⎡ ⎤
⎢ ⎥
⎣ ⎦

1 sub
M I
, nn
M
11 12
21 22
ββ
ββ
⎡ ⎤
⎢ ⎥
⎣ ⎦
, nn
M
10
00
⎡ ⎤
⎢ ⎥
⎣ ⎦

2 sub
M
0
, k k
M
h 20
02h
⎡ ⎤
⎢ ⎥
⎣ ⎦
, k

k
M
20
00
h ⎡ ⎤
⎢ ⎥
⎣ ⎦

3 sub

0 0 0
M
4 sub
M
0 0 0

172
Appendix A.10: The left side mesh points for city problems in the polar coordinate systems

two-dimensional elasti

/2( ) i θ π = line
Boundary Conditions Displacement Traction Mixed (example)
1 sub
N
0 0 0
2 sub
N
0 0 0
1,1
N
11 12
21 22
α α
α α
⎡ ⎤
⎢ ⎥
⎣ ⎦

1,1
N
10
00
⎡ ⎤
⎢ ⎥
⎣ ⎦

,1 kk
N
−
0
0 vr
r − ⎡ ⎤
⎢ ⎥
−
⎣ ⎦

,1 kk
N
−

00
0 vr
⎡ ⎤
⎢ ⎥
−
⎣ ⎦

, kk
N
02h −
20 h
⎡ ⎤
⎢ ⎥
⎣ ⎦

N
10
, kk
20 h
⎡ ⎤
⎢ ⎥
⎣ ⎦

,1 kk
N
+
0 r
0 vr
⎡ ⎤
⎢ ⎥
⎣ ⎦

,1 kk
N
+

00
0 vr
⎡ ⎤
⎢ ⎥
⎣ ⎦

3 sub
N I
, nn
N
11 12
21 22
ββ
ββ
⎡ ⎤
⎢ ⎥
⎣ ⎦

, nn
N
10 ⎡ ⎤
00
⎢ ⎥
⎦

⎣
20
02
h
h
⎡ ⎤
⎢ ⎥
⎣ ⎦

, kk
N
20
00
h ⎡ ⎤
⎢ ⎥
⎣ ⎦

4 sub
N
0
, k
N
k

2
rh θ
11
22
[( )( ) vh r v r h ] h
α
θ
=
−− +

22
12
22
()
[2]
[( )( )]
rvhr
vh r vh h
θ
α
θ
−
=−
−− +
h
r
22
21
22
()
]
[( )( )]
vr h r
vh r vh h
θ
α
θ
+
=+
−− +
[2h
r
22
22
22
[( )( )]
vr h
vh r vh r h
θ
α
θ
=
−− +

2
11
22
[( )( )] vh r vh r h
rh θ
β
θ
=
−− +
22
12
22
()
[2
)]
=−]
[( )(
rvhr
h
vh r vh r h
θ
β
θ
−
−− +

22
21
22
() rhvr θ −
[2]
[( )( )]
h
vh r h r h θ
=− −
−+ −

v
β
22
22
22
[( )( ]
r h
vh r vh r h
θ
β
)
v
θ
=
−

−+

173

174
Appendix B section contains 10 appendices for two-dime onduction problems. The first five appendices are for the
Cartesian coordinate x covers prescribed
temperature, prescribed heat fl
Appendix B.1: The bottom m sh line for two-dimensional heat ms in the Cartesian coordinate systems
Bottom mesh line
nsional heat c
systems. The last five appendices are for the polar coordinate systems. Each appendi
ux, and convection boundary conditions.

e conduction proble

Boundary Conditions Temperature Heat flux Convection
1,1
A
2
h

2
h

2
h
1,1
C 2 − 1 − 2
c
k
khh
−
+

1,2
C 1 1 1
1
P
22
10b
ch gh
TT
kk
22
0
1
flux b
qh
ch gh
T
kk k
ρ
−−

22
10
c
b
c
ch gh h h
TT
kkkhh
ρ
−−
+
ρ
− −

Appendix B.2: The mesh lines for two-dimensional heat conduction problems in the Cartesian coordinate systems
2
nd
~(n-1)
th
mesh lines

Boundary Conditions Any case
, kk
A
2
h

,1 kk
C
−
1
, kk
2 − C
,1 kk
C
+
1
k
P
22
k
ch gh
T
kk
ρ
−

175
Appendix B.3: The to m co in the Cartesian coordinate systems

Top mesh line
p mesh line for two-di ensional nduction problems
Boundary
Conditions
Temperature Heat flux Convection
, nn
A
2
h

2
h

2
h
,1 nn
C
−
1 1 1
, nn
C 2 − 1 − 2
c
k
khh
−
+

n
P
22
0 nt
ch gh
TT
kk
ρ
−−

22
0 flux t
n
qh
ch gh
T
kk k
ρ
−−

22
0
c
nb
c
ch gh h h
TT
kkkhh
ρ
−−
+

Appendix B.4: The left side mesh points for two-dime heat conduction problems in the Cartesian coordinate systems

nsional
0 ( ) x i = line
Boundary Conditions Temperature eat flux Convection H
1 sub
M I 0
, kk
M
c
h
k
2 sub
M
0 I I −
3 sub
M
0 0 0
0 0
4 sub
M
0

176
Appendix B.5: The right side mesh points at c s in the Cartesian coordinate systems

for two-dimensional he

onduction problem
0 ( ) x i = line
Boundary Conditions Temperature Heat flux Convection
1 sub
N
0 0 0
2 sub
N
0 0 0
3 sub
N I 0
, kk
N

c
h
k

4 sub
N
0 I I

Appendix B.6: The bottom mesh line for two-dimensional heat conduction problems in the polar coordinate systems
Bottom mesh line

Boundary Conditions Temperature Heat flux Convection
1,1
A
2
2h

2
2h

2
2h

1,1
C
2
4r −
2
2rrh − −

2
2
(2 )
4
c
rrhk
r
khh
−
−+
+

1,2
2
2rrh

C
2
2rrh +

2
2rrh + +

1
P
1
22 22
22 cr h gr h ρ
2
10
(2 )
r
TrrhT
kk
−− −
1
10
2 22 2
22 (2 )
2
flux r
k k
cr h gr h r rh h ρ −

Tq −−
k
1
22 22 2
22 (2 cr h gr h r rh h h ρ −
10
)
c
r
c
TT
kk khh
−−
+

177
Appendix B.7: The mesh lines for two-dimensional heat conduction problems in the polar coordinate systems

2
nd
~(n-1)
th
mesh lines
Boundary Conditions Any case
, kk
A
2
2h
,1 kk
C
−

2
2rrh −

, kk
C
2
4r −
,1 kk
C
+

2
2rrh +

k
P
22 22
22
k
cr h gr h
T
kk
ρ
−

Appendix B.8: The top mesh line for two-dimensional heat conduction problems in the polar coordinate systems
Top mesh line

Boundary
Conditions
Temp flux Convection erature Heat
, nn
A
2
2h

2
2h

2
2h

,1 nn
C
−

2
2rrh −

2
2rrh −

2
2rrh −

, nn
C
2
4r −
2
2rrh − +

2
2
(2 )
4
rrhk
r
khh
+
−+
+

c
n
P
22 2 2
2
0
22
(2 )
n
nr
cr h gr h
TrrhT
kk
ρ
−− +

22 2 2 2
0
n
flux r
Tq
k k
−
22 (2 )
n
cr h gr h r rh h
k
ρ +
−

22 2 2 2
0
22 (2 )
n
c
nr
c
cr h gr h r rh h h
TT
kk khh
ρ +
−−
+

178
Appendix B.9: The right side mesh points for two-dimensional heat conduction problems in the polar coordinate systems

0 ( ) iii θ = line
Boundary Conditions Temperature eat flux Convection H
1 sub
M I 0

, kk
M

c
h
k
2 sub
M
0 I I −

3 sub
M
0 0 0
4 sub
M
0 0 0

Appendix B.10: The left side mesh points for two-dimensional heat conduction problems in the polar coordinate systems

/2 ( ) i θ π = line
Boundary Conditions Temperature Heat flux Convection
1 sub
N
0 0 0
2 sub
N
0 0 0
3 sub
N I 0

, kk
N

c
h

k

4 sub
N
0 I I

179
Appendix C section contains 14 appendices for plate problems. The first seven appendices are for the Cartesian coordinate systems.
The last seven appendices are for the polar coordinate systems. Each appendix covers clamped, simple supported and free boundary
conditions.

Appendix C.1: The bottom mesh line for plate problems in the Cartesian coordinate systems

Bottom mesh line
Boundary Conditions Clamped Simple supported Free
1,1
A
4
h
4
h

4
h
1,1
B
2
4h −
2
4h −
22
2
2
(42)
vh
vh
L
−−+

1,2
B
2
2h

2
2h

2
(4 ) vh −

1,1
C 7

5

2
2
2
2
vh
L
+

1,2
C 4 − 4 −
2
2
4
vh
L
−−

1,3
C 1 1 2
1
P 10 0
42
bb
A wh θ + −
2
0
10b
2
b
M h
Aw
D
++

23 4
00 0
1 2
2
bb b
MhVh Mvh
A
D DLD
−+ +

4
11
t
z
hh ρ
A wp
D
−+

=

180
Appendix C.2: The second bottom mesh line for plat s

Second bottom mesh line
e problems in the Cartesian coordinate system
Boundary Conditions ped Simple supported Free Clam
2,2
A
4
h
4
h

4
h
2,1
B
2
2h

2
2h

2
(2 ) vh −

2,2
h
B
2 2
4h 4 − −
2
4h −

2,3
B
2
2h

2
2h

2
2h

2,
4 − 4 C
1
− 2 −
2,2
C 6

6

5

2,3
C 4 − 4 − 4 −
2,4
C 1 1 1
2
20b P
A w −

20b
A w −

2
0
2
b
M h
A
D
+
4
2 2
t
z
hh
A wp
ρ
D
−+

=

181
esh lines for plate problems in the C tesian coordinate systems

3
rd
~(n-2)
th
mesh lines
Appendix C.3: The m ar
Boundary Conditions Any case
, kk
A
2
h

,1 kk
B
−

2
2h

, kk
B
2
4h −
,1 kk
B
+

2
2h

,2 kk
C
−
1
,1 kk
C
−
4 −
, kk
C 6

,1 kk
C
+
4 −
,2 kk
C
+
1
k
P
4
t
kz
hh
wp
D
ρ
− +

Appendix C.4: The second top mesh line for plate problems in the Cartesian coordinate systems

182
Second top mesh line
Boundary Conditions Sim Fre Clamped ple supported e

1, 1 nn
A
− −

4
h
4
h

4
h
1, 2 nn
B
− −

2

2h
2
2h

2
2h

1, 1 nn
B
− −

2
4h −
2
4h −
2
4h −

1, nn
B
−

2

2h
2
2h

2
(2 ) vh −

1, 3 nn
C
− −
1 1 1
1, 2 nn
C
−
4 −
−
4 − 4 −
1, 1 nn
C
− −
6

6

5

1, nn
C
−
4 − 4 − 2 −
1 n
P
−
10 nt
A

10 nt
A w
−
−

2
0
1
t
n
M h
A
D
−
+

w −
−
4
11
t
nnz
hh
A wp
D
ρ
−−
=−+

183
Appendix C.5: The top lems in an coordinate systems
Top mesh line
mesh line for plate prob

the Cartesi
Boundary Conditions Clamped Simple supported Free
, nn
A
4
h
4
h

4
h
,1 nn
B
−

2
2h

2
2h

2
(4 ) vh −

, nn
B
2
4h −
2
4h −
22
2
2
(42)
vh
vh
L
−−+

,2 nn
C
−
1 1 2
,1 nn
C
−
4 − 4 −
2
2
4
vh
L
−−

, nn
C 7

5

2
2
2
2
vh
L
+

n
P 00
42
nt t
A wh θ + −
2
0
0
2
t
nt
M h
Aw
D
++

23 4
00 0
2
2
tt t
n
MhVh Mvh
A
D DLD
−+ +

4
t
nnz
hh ρ
A wp
D
−+

=

184
Appendix e pr s s

C.6: The left side mesh points for plat

oblems in the Carte ian coordinate system
0 ( ) x i = line
Boundary Conditions Clamped S ple supported Free im
1,1 1,2
,1 ,
,
,
nn nn
mm
−

mm 0
, kk

2
m v −
11 sub
M I I
,1 ,k −+1
,
kk k
mm v
12 sub
M
0 0 0
1,1 ,
,
n n
mm
22
(1 ) vh −
13 sub
M
0 0
, kk
m
2
h
14 sub
M
0 0 0
1,1 ,
,
nn
mm 2(2 ) / vh L −
21 sub
M
0 0
1,2 , 1
,
nn
mm
−
(2 ) / vh L − −
, kk
m
2(2 ) vh − −
22 sub
M I 0
,1 , 1
,
kk kk
mm
−+
(2 ) vh −
23 sub
M
0 I
1,1 ,
,
nn
mm
3
(2 ) / vhv L −−
24 sub
M
0 0
3
hI

185
Appendix C.7: The right side mesh points for plate problems in the Cartesian coordinate systems

0 ( ) x i = line
Boundary Conditions Clamped Simple supported Free
1,1
0
1,2
,1 ,
,
,
nn nn
nn
nn
−

, k k
n
2v −
31 sub
N I I
,1 , 1
, nn
kk kk −+
v
32 sub
N
0 0 0
1,1 ,
,
nn
nn
22
(1 ) vh −
33 sub
N

0 0
, kk
n
2
h
34 sub
N
0 0 0
1,1 ,
,
nn
nn 2(2 ) / vh L −
41 sub
N
0 0
1,2 , 1
,
nn
nn
−
(2 ) / vh L − −
, kk
n
2(2 ) vh − −
42 sub
N I 0
,1 , 1
,
kk kk
nn
−+
(2 ) vh −
43 sub
N
0 I
1,1 ,
,
nn
nn
3
(2 ) / vhv L −−
44 sub
N
0 0
3
hI

186
Appendix C.8: The bottom mesh line for plate problems in the polar coordinate systems

Bottom mesh line
Boundary Conditions Clamped Simple supported Free
1,1
A
4
2h
4
2h

4
2h
1,1
B
422
88 hrh −
422
88 hrh − 1
a

1,2
B
22 3
42 rh rh −
22 3
42 rh rh −
1
b

1,1
C
43 22
14 2 4 rrh rh −+

43 2
422
2
() Dr vrh −
2( )( )
12 4
rrhr vrh
rrh
−+
+−
1
c

1,2
C
43 22 3
84 2 rrh rh rh −− − +

43 22 3
84 2 rrh rh rh −− − +

d

1
1,3
C
43
22 rrh +

4 3
2rh

2r + 1
e

1
P
(8 Ar −−
0
43 2
4 2 rh r + −
2
10
43
0
2(2 2 )
r
h
hr rh θ
−
+−
3
) rhw
b
0
0
4
4(rr
3 2
10 2
43 22
0 2
)
()
2( )
()
r
r
hr
A w
rvrh
rrhrh
M
Dr vrh
+
−
−
−

0 1 0 1 0 rr r
−
+
10 1
11
A fV g M hM + ++

44 44
11
22
t
z
hr h r h
A wp
D D
ρ
=− +

23 2 2 23 22 22 33
422
1
2
222 2222 32 232
2
(4 2)(6 11 5 222 )
(8 8 )
(2 2 2 )( )
(2 2 )( 4 4 6 5 2 ) ( 8 4 2 )
(2 2 2 )( ) ( )
rh h r h rvh vh r vh rvh v h
ahrh
rrv vh vhr vh
r rh r r v rh vh v h vh r r h rh h vh
rrv vh vhr vh r vh
−+ − + − + −
=+−
−− +
−− + + − + −− + − −
++
−− + − −

−
22 3 3 2 2
22 3
1
22
(4 2 )( )( v 2 ) (2 2 )(2 )
(4 2 )
2 2 ) (2 2 2
rh rhr hv r rhvvh
brhrh
rrv vh vh r rv vh h
+− − − −
=+−−
−− + −− +

(2 ) v
2 2 2 2 23 3 2 2 3 4 3 2 2 2 2 32 2
1
2 2
43 22 3
422
(4 2 )( 4 4 4 2 2 ) (2 2 )(6 7 2 2 )
(2 2 2 )( ) (2 2 2 )( )
2( 8 4 2 )
(12 4 )
()
r rhrvh rvhvh r rvh rhvhr r rh r r vh vh vh
c
r rvvh vhr vh r rvvh vhr vh
rrh rh rhr
rrh
rvh
−+ − + − − + − − − − −
=+
−− + − −− + −
−+ − −
+++
−

v +
232 2 3 223 43 22 2 222
1
2 2
43 22 3
43 22 3
(4 2 )(2 6 2 2 4 4 ) (2 2 )( 4 6 4 2 2 4 )
(2 2 2 )( ) (2 2 2 )( )
()(8 4 2 )
(8 4 2 )
()
r rh r r vh rh vh rvh v h r r r h r r v rvh rv h vh v h
d
rrv vh vhr vh rrv vh vhr vh
rvh r rh rh rh
rrh rh rh
rvh
−+ − + − + − − − + − + + +
=+
−− + − −− + −
+− + − −
−+−−−+
−
22 43
43
1
22
(4 2 )( ) (2 2 )
(2 2 )
(2 2 2 ) (2 2 2 )
r rh r vh r r r h vr
errh
rrv vh vh r rv vh vh
+− −
=− ++
−− + −− +

22 3 2 4 3 3
1
22
(4 2 )(2 2 ) 2(2 2 )
(2 2 2 ) (2 2 2 )
rh rh r vh rh r r h vrh
f
rrv vh vhD rrv vh vhD
+− −
=− +
−− + −− +

22 3 3 4 3 2
1
22
(4 2 ) (2 2 )(2 )
(2 2 2 ) (2 2 2 )
rh rh r r r h v rh
g
rrv vh vhD rr − v vh vhD
+−−
=− +
−− + − +
22 3 3 3 2 2 22 3 4 3 2 2 2 2 4 3 22 3 2
1
2 2
(4 2 )(2 2 2 ) (2 2 )(4 4 2 ) (8 4 2 )
(2 2 2 )() ( r2 2 2 )() ()
r h rh r vh r h r vh r h rvh r r h r r v rvh vh h r r h r h rh rh
h
r rv vh v h r vh D rv v h r vh D r vh D
+ − +− + − − − + + + − +− −
=− +
−− + − −− + − −

vh

187

188
Appendix C.9: The second bottom mesh line for plate problems in the polar coordinate systems
Second bottom mesh line

Boundary Conditions Clamped Simple supported Free
2,2
A
4
2h
4
2h

4
2h
2,1
B
22 3
42 rh rh +
22 3
42 rh rh + 1
i

2,2
B
422
88 hrh −
422
88 hrh −
422
88 hrh −
2,3
B
22 3
42 rh rh −
22 3
42 rh rh −
22 3
42 rh rh −
2,1
C
43 22 3
84 2 rrh rh rh −+ − −
43 22 3
84 2 rrh rh rh −+ − −

1
j

2,2
C
422
12 4 rrh +

422
12 4 rrh +

1
k

2,3
C
43 22 3
84 2 rrh rh rh −− − +
43 22 3
84 2 rrh rh rh −− − +

43 22 3
84 2 rrh rh rh −− − +

2,4
C
43
22 rrh +

43
22 rrh +

43
22 rrh +

2
P 0
43
20
(2 2 )
r
A rrhw −−

0
43
20
(2 2 )
r
A rrhw −−

1
43 2
20
2( )
()
r
rrhrh
AM
Dr vh
−
+
−

44 44
22
22
t
z
hr h r h
A wp
D D
ρ
=− +

22
22 3
1
2( )
(4 2 )
()
rhrvh
irh rh
rvh
−+
=+ +
−

4
43 22 3
1
4( )
(8 4 2 )
()
rhr
j rrh rh rh
rvh
−
=+−+− −
−

3
422
1
2( )( )
(12 4 )
()
rh rvhr
krrh
rvh
−+
=− + +
−

189
Appendix C.10: The mesh lines for plate problems in the polar coordinate systems
3
rd
~(n-2)
th
esh lines

m
Boundary Conditions Any case
, kk
A
2
4h

,1 kk
B
−

22 3
42 rh rh +
, kk
B
422
88 hrh −
,1 kk
B
+

22 3
42 rh rh −
,2 kk
C
−

43
22 rrh −

,1 kk
C
−

43 22 3
84 2 rrh rh rh −+ − −

, kk
C
422
14 4 rrh +

,1 kk
C
+

43 22 3
84 2 rrh rh rh −− − +

,2 kk
C
+

43
22 rrh +

k
P
44 44
22
t
kz
hr h r h
wp
D D
ρ
−+

190
Appendix C.11: The second top mesh line for plate problems in the polar coordinate systems

Second top mesh line
Boundary Conditions Clamped Simple supported Free

1, 1 nn
A
4
2h
4
2h
4
2h
− −

1, 2 nn
B
− −

22 3
42 rh rh +
22 3
42 rh rh +
22 3
42 rh rh +
1, 1 nn
B
− −

422
88 hrh −
422
88 hrh −
422
88 hrh −
1, nn
B
−

22 3
42 rh rh −
22 3
42 rh rh − 2
i

1, 3 nn
C
− −

43
22 rrh −

43
rr 22h −

43
22 rrh −

1, 2 nn
C
− −

43 22 3
84 2 rrh rh rh −+ − −
43 22 3
84 2 rrh rh rh −+ − −

43
rr
22 3
84 2 h rh rh −+ − −

1, 1 nn
C
− −

422
12 4 rrh +

422
12 4 rrh +

2
k

1, nn
C
−

43 22 3
84 2 rrh rh rh −− − +
43 22 3
84 2 rrh rh rh −− − +

2
j

n
P 1
43
10
(2 2 )
n
nr
Ar rhw
+
−
−+

1
43
10 nr
h (2 2 )
n
Ar rw
+
−
−+

43 2
10
2( )
()
n
nr
rrhrh
AM
Dr vh
−
+
+
+
44 44
11
22
t
nn z
hr h r h
A wp
D D
ρ
−−
=−

+
22
22 3
2
2( )
(4 2 )
()
rhrvh
irh rh
rvh
+
=− +
+

4
43 22 3
2
4( )
(8 4 2 )
()
rhr
j rrh rh rh
rvh
+
=+−−− +
+

3
422
2
2( )( )
(12 4 )
()
rvh r hr
krrh
rvh
−+
=− + +
+

191
Appendix C.12: The top mesh line for plate problems in the polar coordinate systems
Top mesh line

Boundary Conditions Clamped Simple supported Free
, nn
A
4
2h
4
2h

4
2h
,1 nn
B
−

22 3
42 rh rh +
22 3
h 42 rh r + 2
b

, nn
B
422
88 hrh −
422
88 hrh − 2
a

,2 nn
C
−

43
22 rrh −

43
22 rrh −

2
e

,1 nn
C
−

43 22 3
84 2 rrh rh rh −+ − −

43 22 3
84 2 rrh rh rh −+ − −

2
d

, nn
C
43 22
14 2 4 rrh rh ++

43 2
422
2
2( )( )
12 4
()
rrhr vrh
rrh
Dr vrh
+−
+−
+

2
c

1
P 0
43 22 3
10
43
0
(8 4 2 )
2(2 2 )
r
b
Ar rh rhrhw
hr rh θ
−− − − +
−+
1
1
43 2
0 nr
11
20 20 20
nn n
nr r r
AfV gM hM
++
+ ++

2
43 22
0 2
4( )
()
2( )
()
n
n
r
rrhr
Aw
rvrh
rrhrh
M
Dr vrh
+
+
+
+
+
+
+
+

44 4 4
22
t
nn z
hr h r h
A wp
D D
ρ
=− +

22 3 3 2 2
22 3
2
22
(4 2 )( )(2 ) (2 2 )(2 )
( 2 )
(2 2 2 ) (2 2 2 )
rh rhrvhv r rhvvh
hrh
rrv vh vh rrv vh vh
−+ − + −
=++−
−+ − − + −
4 br
23 2 2 23 22 22 33
422
2
2
222 2222 32 232
2
(4 2 )(6 11 5 2 2 2 )
(8 8 )
(2 2 2 )
(2 2 )( 4 4 6 5 2 ) ( 8 4 2 )
(2 2 2 )( ) ( )
rh h r h rvh vh r vh rvh v h
ahrh
rrv vh v r vh
r rh r r v rh vh v h vh r r h rh h vh
rrv vh vhr vh r vh
−+ + − − −
=+−
−+ − +
+− + − − + −− − − +
++
−+ − + +
)( h

22 43
43
2
22
(4 2 )( ) (2 2 )
(2 2 )
(2 2 2 ) (2 2 2 )
r rh r vh r r r h vr
errh
rrv vh vh r rv vh vh
−+ +
=− +−
−+ − −+ −

232 2 3 223 43 22 2 222
2
2 2
43 22 3
43 22 3
(4 2 )( 2 6 2 2 4 4 ) (2 2 )( 4 6 4 2 2 4 )
(2 2 2 )( ) (2 2 2 )( )
()(8 4 2 )
(8 4 2 )
()
rrh r rvh rh vh rvh vhr r rh rrv rvh rvh vh vh
d
rrv vh vhr vh rrv vh vhr vh
rvh r rh rh rh
rrh rh rh
rvh
− − − −−− − + − + + − + +
=+
−+ − + −+ − +
−− − − +
−+−+−−
+

2 22 2 23 3 2 2 3 43 22 22 32 2
2
2 2
43 22 3
422
(4 2 )( 4 4 4 2 2 ) (2 2 )(6 7 2 2 )
(2 2 2 )( ) (2 2 2 )( )
2( 8 4 2 )
(12 4 )
()
r rh rvh rvhvh r rvh rhvhr r rh r rv vh vh vh
c
rrv vh vhr vh rrv vh vhr vh
rrh rh rhr
rrh
rvh
−− + + + + + + + − − − −
=+
−+ − + −+ + +
−− − +
+++
+
22 3 2 4 3 3
2
22
(4 2 )(2 2 ) 2(2 2 )
(2 2 2 ) (2 2 2 )
rh rh r vh r h r r h vrh
f
rrv vh vhD rrv vh vhD
−+ +
=−
−+ − −+ −

22 3 3 4 3 2
2
22
(4 2 ) (2 2 )(2 )
(2 2 2 ) (2 2 2 )
rh rh r r r h v rh
g
rrv vh vhD rrv vh vhD
−+−
=− +
−+ − −+ −

192
22 3 3 3 2 2 2 2 3 4 3 2 2 2 2 4 3 2 2 3 2
2
(4 2 )(2 2 2 ) (22 )(44 2 ) (8 4 2 )
(2 2 2 ()
r h rh r vh h r vh r h rvh r r h r r v rvh vh h r r h r h rh rh
h
r rv vh vh r vhD r rv vh vh rvhD rvhD
−− − + − − + − + − + − − − +
=− +
−+ − + −+ − + +

2 2
)( ) (2 2 2 )()
r +

193
Appendix C.13: The right side mesh points for plate problems in the polar coordinate systems

0 ( ) iii θ = line
Boundary Conditions Clamped Simple supported Free
11 sub
M I I
1
A

12 sub
M
0 0 0
1,1
m
22
2(1 )
()
vrh
rvh
−
−

, kk
m
2
2h
13 sub
M
0 0
, nn
m
22
2(1 )
()
vrh
rvh
−
+

14 sub
M
0 0 0
,1 kk
m
−
2
22 vr rh −
, kk
m
2
4vr −
21 sub
M
0
,1 kk
m
+
2
22 vr rh +
1
B

,1 kk
m
+

22
42 2 rrvr rv −+−
, kk
m
22 2 2
84 4 4 rrv h vh −+ + −
22 sub
M I 0
,1 kk
m
−

22
42 2 rrvr rv −−+
23 sub
M
0
2
2hI
1
C

24 sub
M
0 0
3
2hI

194
1
A
:

1,1
m
22
4(1 )
()
vrh
rvh
−−
−
1,2
m
22
4(1 )
()
vrh
rvh
−
−

,1 kk
m
−
2
22 vr rh −
, kk
m
2
4vr −
,1 kk
m
+
2
22 vr rh +
,1 nn
m
−
22
4(1 )
()
vrh
rvh
−−
+
, nn
m
22
4(1 )
()
vrh
rvh
−
+

195
1
B
:

1,1
m
22 3 2 223 2 3 22
22
(4 2 2)( 444 2 2 )
(2 2 2 )( )
r r v r rv r r vh rvh v h rh vh rv h r
rrv vh vhr vhh
θ −+− − + − + − −
−− + −

1,2
m
22 3 2 2 3 2 23
22
2(4 2 2 )( 3 2 2 )
(2 2 2 )( )
r r v r rv r r vh rh vh rvh v h r
rrv vh vhr vhh
θ −+− −+ − + − +
−− + −

1,3
m
22 2
22
(4 2 2 )( )
(2 2 2 )
rrvr rvrvhr
rrv vh vhh
θ −+− −
−
−− +

,3 nn
m
−

22 2
22
(4 2 2 )( )
(2 2 2 )
rrvr rvrvhr
rrv vh vhh
θ −−+ +
−+ −

,1 nn
m
−

22 3 2 2 3 223
22
2(4 2 2 )( 3 2 2 )
(2 2 2 )( )
r r v r rv r r vh rh vh rvh v h r
rrv vh vhr vhh
θ −−+ −− − − − −
−+ − +

, nn
m
22 32 2 23 2 3 22
22
(4 2 2 )( 4 4 4 2 2 )
(2 2 2 )( )
r r v r rv r r vh rvh v h rh vh v rh r
rrv vh vhr vhh
θ −−+ + + + + + −
−+ − +

196
1
C
:

1,1
m
22 2 2 23 22 33 22
2
(4 2 2 )(6 11 5 2 2 2 )
(2 2 2 )( )
r r v r rv r h rvh v h r v h v h rv h
rrv vh vhr vhr
θ −+− − + − − +
−
−− + −

1,2
m
22
2
(4 2 2 )( )(2 )
(2 2 2 )
rrvr rvrvh v
rrv vh vh
θ −+− − −
−− +

,1 nn
m
−

22
2
(4 2 2 )( )(2 )
(2 2 2 )
rrvr rvrvh v
rrv vh vh
θ −−+ + −
−+ −

, nn
m
22 2 2 23 22 33 22
2
(4 2 2 )(6 11 5 2 2 2 )
(2 2 2 )( )
r r v r rv r h rvh v h rv h v h r v h
rrv vh vhr vhr
θ −−+ + + − − −
−+ − +

197
Appendix C.14: The left side mesh points for plate problems in the polar coordinate systems

0 ( ) iii θ = line
Boundary Conditions Clamped Simple supported Free
31 sub
N I I
2
A

32 sub
N
0 0 0
1,1
n
22
2(1 )
()
vrh
rvh
−
−

, kk
n
2
2h
33 sub
N
0 0
, nn
n
22
2(1 )
()
vrh
rvh
−
+

34 sub
N
0 0 0
,1 kk
n
−
2
22 vr rh −
, kk
n
2
4vr −
41 sub
N
0
,1 kk
n
+
2
22 vr rh +
2
B

,1 kk
n
+

22
42 2 rrvr rv −+−
, kk
n
22 2 2
84 4 4 rrv h vh −+ + −
42 sub
N I 0
,1 kk
n
−

22
42 2 rrvr rv −−+
43 sub
N
0
2
2hI
2
C

44 sub
N
0 0
3
2hI

198
2
A
:

1,1
n
22
4(1 )
()
vrh
rvh
−−
−
1,2
n
22
4(1 )
()
vrh
rvh
−
−

,1 kk
n
−
2
22 vr rh −
, kk
n
2
4vr −
,1 kk
n
+
2
22 vr rh +
,1 kk
n
−
22
4(1 )
()
vrh
rvh
−−
+
, nn
n
22
4(1 )
()
vrh
rvh
−
+

199
2
B
:

1,1
n
22 3 2 223 2 3 22
22
(4 2 2)( 444 2 2 )
(2 2 2 )( )
r rvr rvr rvh rvhvhrhvh rvhr
rrv vh vhr vhh
θ −+− − + − + − −
−− + −

1,2
n
22 3 2 2 3 2 23
22
2(4 2 2 )( 3 2 2 )
(2 2 2 )( )
r r v r rv r r vh rh vh rvh v h r
rrv vh vhr vhh
θ −+− −+ − + − +
−− + −

1,3
n
22 2
22
(4 2 2 )( )
(2 2 2 )
rrvr rvrvhr
rrv vh vhh
θ −+− −
−
−− +

,3 nn
n
−

22 2
22
(4 2 2 )( )
(2 2 2 )
rrvr rvrvhr
rrv vh vhh
θ −−+ +
−+ −

,1 nn
n
−

22 3 2 2 3 223
22
2(4 2 2 )( 3 2 2 )
(2 2 2 )( )
r r v r rv r r vh rh vh rvh v h r
rrv vh vhr vhh
θ − −+ − − −−− −
−+ − +

, nn
n
22 32 2 23 2 3 22
22
(4 2 2 )( 4 4 4 2 2 )
(2 2 2 )( )
r r v r rv r r vh rvh v h rh vh v rh r
rrv vh vhr vhh
θ −−+ + + + + + −
−+ − +

200
2
C
:

1,1
n

22 2 2 23 22 33 22
2
(4 2 2 )(6 11 5 2 2 2 )
(2 2 2 )( )
r r v r rv r h rvh v h r v h v h rv h
rrv vh vhr vhr
θ − +− − +− −+
−
−− + −

1,2
n
22
2
(4 2 2 )( )(2 )
(2 2 2 )
rrvr rvrvh v
rrv vh vh
θ −+− − −
−− +

,1 nn
n
−

22
2
(4 2 2 )( )(2 )
(2 2 2 )
rrvr rvrvh v
rrv vh vh
θ −−+ + −
−+ −

, nn
n
22 2 2 23 22 33 22
2
(4 2 2 )(6 11 5 2 2 2 )
(2 2 2 )( )
r r v r rv r h rvh v h rv h v h r v h
rrv vh vhr vhr
θ −−+ + + − − −
−+ − +

Abstract (if available)

Abstract This dissertation presents a novel developed semi-analytic mathematic scheme, the finite difference distributed transfer function method (FDDTFM), succeeds both static/steady-state and dynamic/unsteady-state analysis on the two-dimensional elasticity, heat conduction, and plate problems in the Cartesian and the polar coordinate systems. The scheme uses the finite difference method (FDM) along the y/r direction and the distributed transfer function method (DTFM) along the x/theta direction. The FDDTFM doesn't need to re-derive equations because of changing boundary conditions

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University of Southern California Dissertations and Theses

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Asset Metadata

Creator Yang, Yau-Bin (author)

Core Title Static and dynamic analysis of two-dimensional elastic continua by combined finite difference with distributed transfer function method

School Viterbi School of Engineering

Degree Doctor of Philosophy

Degree Program Aerospace

Publication Date 09/10/2007

Defense Date 08/19/2007

Publisher University of Southern California (original), University of Southern California. Libraries (digital)

Tag Cartesian coordinate system,distributed transfer function method,elasticity,finite difference method,heat conduction,OAI-PMH Harvest,plate theory,polar coordinate system,semi-analytic mathematic scheme,static and dynamic,steady-state and unsteady-state,two-dimensional elastic continua

Language English

Advisor Yang, Bingen (committee chair), Flashner, Henryk (committee member), Lee, Vincent W. (committee member), Sadhal, Satwindar S. (committee member), Shiflett, Geoffrey R. (committee member)

Creator Email yaubinya@usc.edu

Permanent Link (DOI) https://doi.org/10.25549/usctheses-m812

Unique identifier UC1338130

Identifier etd-Yang-20070910 (filename),usctheses-m40 (legacy collection record id),usctheses-c127-601404 (legacy record id),usctheses-m812 (legacy record id)

Legacy Identifier etd-Yang-20070910.pdf

Dmrecord 601404

Document Type Dissertation

Rights Yang, Yau-Bin

Type texts

Source University of Southern California (contributing entity), University of Southern California Dissertations and Theses (collection)

Repository Name Libraries, University of Southern California

Repository Location Los Angeles, California

Repository Email cisadmin@lib.usc.edu