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Chapter 2 The construction of the heat equation and methods which have been used to solve it 2.1 The heat equation We will be using the heat equation as the basis of our work and so we begin by giving a derivation in its most general form. The construction of the heat equation is to be found in many publications. Examples include Carslaw and Jaeger (1959), Spiegel (1959), Weinberger (1965), Kreider et al. (1966) and Crank (1979). We consider a material, with density ρ, and specific heat c, which oc- cupies a region of space, V , and which is bounded by a surface S. If the temperature of the material at any point in V is u(r,t), where r is the usual position vector, then the total heat energy contained in the solid is � ρcu dV . V Heat may only enter or leave the region by flowing across the boundary S. We consider the heat flux vector, q, which represents the rate of heat flow 4
at a point per unit area. If we have a small element of the surface area dS with outward unit normal vector ˆn, then the rate of heat flow outward through this element of surface area is q.ˆn dS and so over the whole surface the total rate of heat flow outwards is � S q.ˆn dS According to the law of conservation of energy, the rate at which the total heat energy within the region V changes must balance with the amount of heat crossing the boundary. Therefore � � d ρcu dV + q.ˆndS = 0 dt V S because, if the flux of heat out of the region is positive, then the total energy inside the region will decrease, so its derivative will be negative. We now apply the divergence theorem to the flux integral and take the time derivative inside the volume integral to obtain or � V � � ∂ (ρcu) dV + div q dV = 0 ∂t V V � � ∂ (ρcu) + div q dV = 0 ∂t This expression has to be true for any volume V, which means that the integrand must be zero everywhere. Therefore we must have ∂ (ρcu) + divq = 0 ∂t From Fourier’s law of heat flow we know that q = −Kgradu (2.1) where K is the thermal conductivity, which is not necessarily constant, and div (Kgradu) = ∂ ∂t (ρcu) 5
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 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: system. In 1998 the Department was
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
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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
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Chapter 8 The use of multiple proce
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In considering parallel computing m
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The issue with shared memory system
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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
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not occur when only one process is
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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
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simple. 9.2 Research objectives Our
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Laplace transform, new complexities
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orthogonal flow lines. To implement
Page 203 and 204:
Crank J (1957) Two methods for the
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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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