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Tables of Contents for The Green Element Method
Chapter/Section Title
Page #
Page Count
Preface
xiii
Acknowledgments
xv
Preliminaries
1
14
Introduction
1
3
Motivation for the Green Element Method
4
5
Elementary Matrix Algebra
9
3
References
12
3
Linear Laplace/Poisson Equation
15
36
Introduction
15
1
Derivation of Linear Laplace/Poisson Equation
15
2
Boundary Conditions
17
1
Integral Equations
18
8
Conventional Procedure of Implementing the Boundary Element Method
26
1
Solution Procedure of the Green Element Method
27
6
Choice of the Value of k
29
1
Assembling the Global Matrix
30
3
Comparison of Boundary Element and Green Element Formulations
33
1
Evaluation of the Line Integral of the Recharge Term
34
5
Numerical Examples
39
8
Remarks
47
1
Exercises
48
1
References
49
2
Nonlinear Laplace/Poisson Equation
51
46
Introduction
51
1
Derivation of Differential Equation (Heat Conduction Example)
51
2
Integral Representation
53
1
First Green Element Model (GESNH-1)
54
2
Interpolation
56
2
One-Dimensional Elements
58
4
Linear Elements
58
3
Quadratic Elements
61
1
Second Green Elements Model (GESNH-2)
62
5
Third Green Element Model (GESNH-3)
67
2
Examples on Linear Heterogeneous Laplace/Poisson Problems
69
8
Implementation of GESNH-1 on a 3-Element Computational Region
71
1
Implementation of GESNH-2 on a 3-element Computational Region
72
1
Implementation of GESNH-3 on a 3-Element Computational Region
73
4
Nonlinear Solution Strategies
77
5
Linear Iteration or Picard scheme
77
2
The Newton-Raphson Solution Algorithm
79
3
Applications of Picard and Newton-Raphson Algorithms in Green Element Calculations
82
2
Example 2 --- Nonlinear Example
84
2
Applications to Structural Analysis of Beams
86
8
Example 3
88
3
Example 4
91
3
Remarks
94
1
Exercises
94
2
References
96
1
Helmholtz Equation
97
20
Introduction
97
1
Derivation of Helmholtz Equation
97
2
Two Green Element Models
99
9
Model 1
99
2
Model 2
101
7
Numerical Experiments to Establish Suitable Range of Values for the Penalty Parameter
108
2
Hand Computations
110
2
Computer Simulations of the Three Helmholtz Examples
112
1
Remarks
113
1
Exercises
114
3
Transient Diffusion
117
36
Introduction
117
1
Derivation of the Transient Diffusion Equation
117
3
Boundary and Initial Conditions
120
1
Quasi-Steady Green Element Model (QSGE)
120
4
Transient Green Element Model (TGE)
124
7
Stability Characteristics of GEM for the Diffusion Equation
131
9
Numerical Examples of Transient Linear Diffusion
140
2
Nonlinear Diffusion
142
4
Numerical Examples on Transient Nonlinear Diffusion
146
3
Remarks
149
1
Exercises
150
1
References
151
2
Transport Equation
153
42
Introduction
153
1
Model 1: Quasi-Steady Green Element Model (QSGE)
154
4
Mode 2 (TGE Model)
158
3
Model 3 (ADGE Model)
161
9
Stability Characteristics of GEM for the Transport Equation
170
11
Numerical Examples
181
10
Remarks
191
2
Exercises
193
1
References
194
1
Burgers Equation
195
22
Introduction
195
2
Model 1 for the Burgers Equation
197
3
Model 2 for the Burgers Equation
200
2
Model 3 for the Burgers Equation
202
2
Numerical Examples
204
2
Numerical Experiments and Results
206
8
Remarks
214
1
Exercises
215
1
References
215
2
Unsaturated Flow (Richards Equation)
217
14
Introduction
217
2
Derivation of the Unsaturated Flow Equation
219
3
Green Element Formulation
222
3
Numerical Examples
225
5
References
230
1
Higher-Order Elements
231
20
Introduction
231
1
Green Element Equations with Hermitian Interpolation Polynomials
232
8
Hermitian GEM for the Transport Equation
233
2
Hermitian GEM for Burgers Equation
235
2
Hermitian GEM for the Unsaturated Flow Equation
237
3
Numerical Stability Characteristics of Hermitian GEM for the Transport Equation
240
3
Numerical Calculations
243
7
Examples of Contaminant Transport
243
1
Examples of Momentum Transport (Burgers Equation)
244
3
Examples of Unsaturated Flow
247
3
References
250
1
Steady Two-Dimensional Problems
251
38
Introduction
251
1
Steady Second-Order Differential Equation
251
1
Boundary Conditions
252
1
Integral Equations
253
2
Green Element Models
255
5
Polygonal Elements and their Interpolation Functions
260
3
Bilinear Rectangular Element
260
1
Linear Triangular Element
261
2
Boundary Integrations
263
2
Domain Integrations
265
6
Numerical Examples
271
17
Remarks
288
1
References
288
1
Unsteady Two-Dimensional Problems
289
26
Introduction
289
1
Flow Equation
289
1
Green Element Modeling
290
9
Quasi-Steady Models
290
5
Transient Model
295
4
Numerical Examples
299
12
Remarks
311
2
References
313
2
Further Considerations
315
18
Introduction
315
1
Quadratic Rectangular and Triangular Elements
315
3
Quadratic Rectangular Element
316
1
Quadratic Triangular Elements
317
1
Domain Integrations For Quadratic Elements
318
6
Simulation of Example With Quadratic Elements
324
1
Isoparametric Elements
324
5
Three-Dimensional Problems
329
2
References
331
2
APPENDIX A
333
2
A.1 Capabilities of the Program GEMLN1D
333
1
A.2 User Manual of the Executable Program GEMLN1D
333
1
A.3 Hint to Solution to some of the Exercises
334
1
APPENDIX B
335
4
B.1 Element Matrices for Linear Rectangular Elements
335
2
B.2 Element Matrices for Linear Triangular Elements
337
2
APPENDIX C
339
10
C.1 Element Matrices for Quadratic Rectangular Elements
339
5
C.2 Element Matrices for Quadratic Triangular Elements
344
5
Author Index
349
2
Subject Index
351