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List of Figures 2.1 Grid for using the finite difference method . . . . . . . . . . . 12 2.2 Typical grid for the finite element method . . . . . . . . . . 15 2.3 Diagram showing the discretisation of the boundary into ele- ments and the collocation of a typical base node with a target element for the boundary element method . . . . . . . . . . . 18 2.4 Region of geometry for the method of fundamental solutions . 24 3.1 The movement of the isotherms with time in example 3.1 . . 43 3.2 Temperature plotted as a function of position in example 3.2 for ˜t = 0.05 . . . . . . . . . . . . . . . . . . . . . . . . . . . . 44 3.3 The positions of the isotherms for 0 � ˜x � 5 plotted as a function of time for example 3.2 . . . . . . . . . . . . . . . . . 45 3.4 The graph of f1(˜x1, ˜t) for ˜t = 0.1 in example 3.2 . . . . . . . . 46 3.5 The graph of f2(˜x2, ˜t) for ˜t = 0.1 in example 3.2 . . . . . . . . 47 3.6 The graphs of ũ(˜x, ˜t) = f1(˜x1, ˜t) + f2(˜x2, ˜t) and the analytic solution at ˜t = 0.1 in example 3.2 . . . . . . . . . . . . . . . . 47 3.7 The graph of f1(˜x1, ˜t) for ˜t = 0.5 in example 3.2 . . . . . . . . 48 3.8 The graph of f2(˜x2, ˜t) for ˜t = 0.5 in example 3.2 . . . . . . . . 48 3.9 The graphs of ũ(˜x, ˜t) = f1(˜x1, ˜t) + f2(˜x2, ˜t) and the analytic solution at ˜t = 0.5 in example 3.2 . . . . . . . . . . . . . . . . 49 3.10 The positions of isotherm 2 for linear variation in α in example 3.4 . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 55 x
3.11 The positions of isotherm 8 for linear variation in α in example 3.4 . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 55 4.1 Comparison of the Laplace transform isotherm migration method with the analytic solution for isotherms 2 and 8 in example 4.1 72 4.2 Position as a function of temperature, u, for times t = 0.1,0.2,0.3 and 1.5 when α = 1 + u in example 4.2 . . . . . . . . . . . . . 74 4.3 Position as a function of temperature, u, for times t = 0.1,0.2,0.3 and 1.5 when the average value of α is used to start the problem in example 4.2 . . . . . . . . . . . . . . . . . . . . . . . . 76 4.4 Position as a function of temperature, u, for times t = 0.05,0.15,0.25 and 1.45 when α = exp(−u) in example 4.3 . . . . . . . . . . 78 5.1 Diagram showing the problem described by Crank and Phale (1973) . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 82 5.2 The position of the melting front calculated using the Laplace transform without updating (LT1) and with updating (LT2) in example 5.1 . . . . . . . . . . . . . . . . . . . . . . . . . . 86 5.3 The position of isotherm with temperature u = 8 calculated using the Laplace transform without updating (LT1) and with updating (LT2)in example 5.1 . . . . . . . . . . . . . . . . . . 86 5.4 The position of the melting front and isotherm with temperature u = −8 with increasing time in example 5.2 . . . . . . . 93 5.5 Diagram showing the approximate positions of the isotherms in the case when there is convection across the wall at ˜x = 0 . 96 5.6 Comparison of the position of the melting front for the cases when the wall temperature is held steady and when there is convection across the wall, Bi = 5, in example 5.3 . . . . . . 99 xi
Page 1 and 2: Aspects of the Laplace transform is
Page 3 and 4: Acknowledgements It is my greatest
Page 5 and 6: Abstract There are many different m
Page 7 and 8: Contents 1 Introduction 1 2 The con
Page 9 and 10: 4.7.1 Contribution . . . . . . . .
Page 11: 9.2.3 To extend the use of Laplace
Page 15 and 16: 7.10 Diagram showing the position o
Page 17 and 18: List of Tables 3.1 The position of
Page 19 and 20: 6.1 The positions of the freezing f
Page 21 and 22: u temperature ũ dimensionless u co
Page 23 and 24: Chapter 1 Introduction We begin our
Page 25 and 26: system. In 1998 the Department was
Page 27 and 28: at a point per unit area. If we hav
Page 29 and 30: where G(x,y) is some function indep
Page 31 and 32: condition may vary from one point o
Page 33 and 34: We can find the solution in differe
Page 35 and 36: this is the initial condition. The
Page 37 and 38: Figure 2.2: Typical grid for the fi
Page 39 and 40: fundamental solution which satisifi
Page 41 and 42: tending the technique to non-homoge
Page 43 and 44: 2.2.7 The method of separation of v
Page 45 and 46: to a set of linear equations of the
Page 47 and 48: labelled Qk, where k = 1,2,...,n
Page 49 and 50: in the case of melting ice (Crank a
Page 51 and 52: and the moving boundary differs. In
Page 53 and 54: 2.3.5 Fixed-domain methods In certa
Page 55 and 56: 2.4 Summary of Chapter 2 In this ch
Page 57 and 58: e easily evaluated, topological cha
Page 59 and 60: We wish to write the heat flow equa
Page 61 and 62: Equations (3.8), (3.9), (3.10) and
Page 63 and 64:
In this example we consider the cas
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and ˜x 0.8 0.7 0.6 0.5 0.4 0.3 0.2
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˜x 0.5 0.4 0.3 0.2 0.1 Isotherm 0
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w 5 4.5 4 3.5 3 2.5 2 1.5 1 0.5 0 0
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ũ 10 9 8 7 6 5 0 0.2 0.4 0.6 0.8 1
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Table 3.5: Effect of errors in star
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Table 3.7: Effect of errors in star
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˜x 0.8 0.7 0.6 0.5 0.4 0.3 0.2 0.1
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it is much more straightforward to
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Chapter 4 The Laplace transform iso
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available in tables. We do have an
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N� A = − 3. Gauss quadrature, w
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The value of M is chosen by the use
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are found where wj are the Stehfest
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where h is the difference in the va
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Table 4.3: Comparison of the result
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α = 1 + u. For the steady-state so
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Table 4.5: The positions of the iso
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To start this problem, we use an ac
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4.7 Summary of Chapter 4 In this ch
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differences, and we shall show how
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˜x = 0, ũ = ũL, ˜t � 0 As in
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However, we find that doing this ca
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front and isotherm 8 respectively u
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straightforward and might require o
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freezing front is given by x0 = 2φ
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x 1 0.8 0.6 0.4 0.2 Freezing front
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ũ(˜x, ˜t) = 0, ˜x = ˜x0, ˜t
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so that when X = 0 Eliminating ũw
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˜x 1 0.8 0.6 0.4 0.2 Steady Convec
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˜x 1 0.8 0.6 0.4 0.2 Steady Convec
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Page 127 and 128:
∂u ∂y = 0 y = 0 0 � x � 1 t
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6.3 The finite difference form We e
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Results We see from table 6.1 that
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where t0 is the initial time at whi
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x 1 0.8 0.6 0.4 0.2 Freezing front
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grid line x = xM = 1 for all t > 0.
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7.2 Solidification of a square pris
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which is our previous equation (6.8
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Figure 7.2: Diagram showing possibl
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Poots (1962) defines a function �
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y 1 0.8 0.6 0.4 0.2 t=0.05 t=0.1 t=
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so that we have ¯y (n) = y (t0) λ
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y 0.9 0.8 0.7 0.6 0.5 0.4 0.3 0.2 0
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y 0.7 0.6 0.5 0.4 0.3 0.2 0.1 0 0 0
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Table 7.3: Values of the y co-ordin
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y 1 0.8 0.6 0.4 0.2 0 0 0.2 0.4 0.6
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y 1 0.8 0.6 0.4 0.2 0 0 0.2 0.4 0.6
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y 0.4 0.35 0.3 0.25 0.2 0.15 0.1 0.
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espaced using a given algorithm. We
Page 167 and 168:
Chapter 8 The use of multiple proce
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In considering parallel computing m
Page 171 and 172:
The issue with shared memory system
Page 173 and 174:
Figure 8.3: Diagram showing a distr
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are several MPI implementations ava
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communication in the program adds e
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Sp 4 3.5 3 2.5 2 1.5 1 1 1.5 2 2.5
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work. Because the calculation invol
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Table 8.3: Speed-up when allocating
Page 185 and 186:
not occur when only one process is
Page 187 and 188:
Therefore, as in examples 8.1 and 8
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upon which we focus. We mention the
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heating of a rod at zero degrees, n
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described by an equation involving
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grid points. An example, described
Page 197 and 198:
simple. 9.2 Research objectives Our
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Laplace transform, new complexities
Page 201 and 202:
orthogonal flow lines. To implement
Page 203 and 204:
Crank J (1957) Two methods for the
Page 205 and 206:
Davies A J and Mushtaq J (1998) Par
Page 207 and 208:
Kreider D L, Kuller R G, Ostberg D
Page 209 and 210:
Price R H and Slack M R (1954) The
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