Page 25 - Fundamentals of Computational Geoscience Numerical Methods and Algorithms

2.2 Finite Element Formulation of the Problem 11
2.2 Finite Element Formulation of the Problem

By considering the dimensionless velocity, pressure and temperature as basic vari-
ables, Eqs. (2.10), (2.11), (2.12) and (2.13) can be discretized using the conventional
finite element method (Zienkiewicz 1977, Zhao et al. 1997a). For a typical 4-node
quadrilateral element, the velocity, pressure and temperature fields at the elemental
level can be expressed as

T
e
∗
∗
∗
u (x , y ) = ϕ U , (2.15)
e
T
∗
∗
∗
v (x , y ) = ϕ V , (2.16)
T
e
∗
∗
∗
P (x , y ) = Ψ P , (2.17)
T
e
∗
∗
∗
T (x , y ) = ϕ T , (2.18)
e
e
e
e
where U , V , P and T are the column vectors of the nodal velocity, excess pres-
sure and temperature of the element; ϕ is the column vector of the interpolation
functions for the dimensionless velocity and temperature fields within the element;
Ψ is the column vector of the interpolation functions for the excess pressure within
the element. For the 4-node quadrilateral element, it is assumed that ϕ is identical
to Ψ in the following numerical analysis.
The global coordinate components within the element can be defined as
T
T
∗
x = N X, y = N Y, (2.19)
∗
where X and Y are the column vectors of nodal coordinate components in the x and
y directions of the global coordinate system respectively; N is the column vector of
the coordinate mapping function of the element. Based on the isoparametric element
concept, the following relationships exist:
N(ξ, η) = ϕ(ξ, η) = Ψ(ξ, η), (2.20)
where ξ and η are the local coordinate components of the element.
Using the Galerkin weighted-residual method, Eqs. (2.10), (2.11), (2.12) and
(2.13) can be expressed, with consideration of Eqs. (2.15), (2.16), (2.17) and (2.18),
as follows:
∂ϕ e ∂ϕ e
T T
Ψ U dA + Ψ V dA = 0, (2.21)
A ∂x ∗ A ∂y ∗
T
∂Ψ
T e ∗ e ∗ T e
ϕϕ U dA + ϕK x P dA + ϕK Raϕ T e 1 dA = 0, (2.22)
x
A A ∂x ∗ A