Page 27 - Fundamentals of Computational Geoscience Numerical Methods and Algorithms

2.2 Finite Element Formulation of the Problem 13

T
∂ϕ ∂ϕ ∂ϕ
e e e
∗
∗
D (v ) = ϕν ∗ dA, L = λ ∗ ey dA, F = σ ϕdS,
y
y
y
y
A ∂x ∗ A ∂y ∗ ∂y ∗ S
(2.31)

e e ∗ e ∗ e e e ∗
E = D (u ) + D (v ) + L + L , G =− q ϕdS, (2.32)
x
y
x
y
S
H

e T ∗ ∗ K h ρ f 0 c p
M = ϕϕ dA, q = q, σ = σ, (2.33)
A ΔT λ e0 μλ e0
where ϕ is the shape function vector for the temperature and velocity components
of the element; Ψ is the shape function vector for the pressure of the element; σ and
q are the stress and heat flux on the boundary of the element; A and S are the area
and boundary length of the element.
It is noted that since the full nonlinear term of the energy equation in the Horton-
e
Rogers-Lapwood problem is considered in the finite element analysis, matrix E is
dependent on the velocity components of the element. Thus, a prediction for the ini-
tial velocities of an element is needed to have this matrix evaluated. This is the main
motivation for proposing a progressive asymptotic approach procedure in the next
section.
From the penalty finite element approach (Zienkiewicz 1977), the following
equation exists:
e
e
e
e
e
C U + C V =−εM p P . (2.34)
y
x
Equation (2.34) can be rewritten as
1
e −1 e e e e
P =− M (C U + C V ). (2.35)
x
y
p
ε
Substituting Eq. (2.35) into Eq. (2.27) yields the following equation in the ele-
mental level:
e e e e
Q −B U F = F , (2.36)
0 E e T e G e
where
1
e
e
e −1
e T
e
Q = M + A (M ) (C ) , (2.37)
p
ε
e e e
e M 0 e U e F x
M = e , U = e , F = e , (2.38)
F
0M V F y
B A C
e e e
e x e x e x
B = e , A = e , C = e , (2.39)
B A C
y y y