Civil Engineering Reference
In-Depth Information
Writing Equation 6.84 in matrix form,
∂
N
i
∂
∂
N
i
∂
∂
x
∂
y
x
r
∂
r
∂
r
=
i
=
1, 4
(6.85)
∂
N
i
∂
∂
N
i
∂
∂
x
∂
y
y
s
∂
s
∂
s
we observe that the
2
×
1
vector on the left-hand side is known, since the inter-
polation functions are expressed explicitly in the natural coordinates. Similarly,
the terms in the
2
×
2
coefficient matrix on the right-hand side are known via
Equation 6.80. The latter, known as the
Jacobian matrix,
denoted [J], is given
by
i
=
1
∂
i
=
1
∂
4
N
i
∂
4
N
i
∂
∂
x
∂
y
x
i
y
i
=
r
r
∂
r
∂
r
[J]
=
(6.86)
∂
x
∂
y
i
=
1
∂
4
N
i
∂
i
=
1
4
∂
N
i
∂
x
i
y
i
∂
∂
s
s
s
s
If the inverse of the Jacobian matrix can be determined, Equation 6.85 can be
solved for the partial derivatives of the interpolation functions with respect to the
global coordinates to obtain
∂
N
i
∂
∂
N
i
∂
∂
N
i
∂
I
11
x
I
12
r
r
[J]
−
1
=
=
i
=
1, 4
(6.87)
∂
N
i
∂
∂
N
i
∂
I
21
I
22
∂
N
i
∂
y
s
s
with the terms of the inverse of the Jacobian matrix denoted
I
ij
for convenience.
Equation 6.87 can be used to obtain the partial derivatives of the field variable
with respect to the global coordinates, as required in discretizing a governing dif-
ferential equation by the finite element method. In addition, the derivatives are
required in computing the “secondary” variables, including strain (then stress) in
structural problems and heat flux in heat transfer. These and other problems are
illustrated in subsequent chapters.
As we also know, various integrations are required to obtain element stiff-
ness matrices and load vectors. For example, in computing the terms of the con-
ductance matrix for two-dimensional heat transfer elements, integrals of the
form
∂
d
A
N
i
∂
∂
N
j
∂
x
x
A
are encountered, and the integration is to be performed over the area of the
element in global coordinates. However, for an isoparametric element such as
the quadrilateral being discussed, the interpolation functions are in terms of the
parent element coordinates. Hence, it is necessary to transform such integrals to
