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Glossary The terms included in this glossary refer to items defined in the Laplace transform isotherm migration method. For clarity we do not include sym- bols used in chapter 2, the review of other methods. Bi dimensionless Biot number, hclc/K c specific heat f(t) function of time ¯f(λ) transformed function of time g(x,y) shape function of boundary condition h difference in temperatures of adjacent isotherms hc heat transfer coefficient K thermal conductivity Kliq Ksol l length lc thermal conductivity in the liquid phase thermal conductivity in the solid phase characteristic length, the volume of a body divided by its surface area L latent heat of fusion L [f(t)] Laplace transform M number of Stehfest parameters p number of processors s dimensionless Stefan number, L/c Sp speed-up t time ˜t dimensionless t co-ordinate t0 initial time T specific value of t at which a solution is required xviii
u temperature ũ dimensionless u co-ordinate u0 ũ0 uf initial temperature dimensionless u0 co-ordinate fusion temperature uF(x,y,t) = 0 contour on freezing front ui uL uliq ũL uR ũR usol Ui Ui,j isotherm of specific temperature temperature of left boundary temperature in the liquid phase dimensionless temperature of left boundary temperature of right boundary dimensionless temperature of right boundary temperature in the solid phase approximation to u at grid point (xi) approximation to u at grid point (xi,tj) V constant temperature of liquid phase uv uw ũw steady temperature outside wall at xL temperature of wall at xL dimensionless temperature of wall at xL v temperature function f1(˜x1, ˜t) v1 v ′ 1 v2 v ′ 2 temperature of ice transformed temperature of ice temperature of water transformed temperature of water w temperature function f2(˜x2, ˜t) wj Stehfest weight ˜x dimensionless x co-ordinate x0 position of isotherm with temperature u0 xix
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 and 12: 9.2.3 To extend the use of Laplace
Page 13 and 14: 3.11 The positions of isotherm 8 fo
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: 6.1 The positions of the freezing f
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
Page 65 and 66: and ˜x 0.8 0.7 0.6 0.5 0.4 0.3 0.2
Page 67 and 68: ˜x 0.5 0.4 0.3 0.2 0.1 Isotherm 0
Page 69 and 70: w 5 4.5 4 3.5 3 2.5 2 1.5 1 0.5 0 0
Page 71 and 72:
ũ 10 9 8 7 6 5 0 0.2 0.4 0.6 0.8 1
Page 73 and 74:
Table 3.5: Effect of errors in star
Page 75 and 76:
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
Page 81 and 82:
Chapter 4 The Laplace transform iso
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available in tables. We do have an
Page 85 and 86:
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
Page 101 and 102:
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
Page 107 and 108:
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
Page 113 and 114:
freezing front is given by x0 = 2φ
Page 115 and 116:
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
Page 125 and 126:
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
Page 131 and 132:
Results We see from table 6.1 that
Page 133 and 134:
where t0 is the initial time at whi
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x 1 0.8 0.6 0.4 0.2 Freezing front
Page 137 and 138:
Page 139 and 140:
grid line x = xM = 1 for all t > 0.
Page 141 and 142:
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
Page 147 and 148:
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
Page 157 and 158:
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
Page 163 and 164:
y 0.4 0.35 0.3 0.25 0.2 0.15 0.1 0.
Page 165 and 166:
espaced using a given algorithm. We
Page 167 and 168:
Chapter 8 The use of multiple proce
Page 169 and 170:
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
Page 175 and 176:
are several MPI implementations ava
Page 177 and 178:
communication in the program adds e
Page 179 and 180:
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
Page 183 and 184:
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
Page 189 and 190:
upon which we focus. We mention the
Page 191 and 192:
heating of a rod at zero degrees, n
Page 193 and 194:
described by an equation involving
Page 195 and 196:
grid points. An example, described
Page 197 and 198:
simple. 9.2 Research objectives Our
Page 199 and 200:
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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