Civil Engineering Reference
In-Depth Information
If the values of the field variable are computed only at nodes, how are values
obtained at other points within a finite element? The answer contains the crux of
the finite element method: The values of the field variable computed at the nodes
are used to approximate the values at nonnodal points (that is, in the element
interior) by
interpolation
of the nodal values. For the three-node triangle exam-
ple, the nodes are all exterior and, at any other point within the element, the field
variable is described by the approximate relation
3
(1.1)
where
1
,
2
, and
3
are the values of the field variable at the nodes, and
N
1
,
N
2
,
and
N
3
are the
interpolation functions,
also known as
shape functions
or
blend-
ing functions
. In the finite element approach, the nodal values of the field vari-
able are treated as unknown
constants
that are to be determined. The interpola-
tion functions are most often polynomial forms of the independent variables,
derived to satisfy certain required conditions at the nodes. These conditions are
discussed in detail in subsequent chapters. The major point to be made here is
that the interpolation functions are predetermined,
known
functions of the inde-
pendent variables; and these functions describe the variation of the field variable
within the finite element.
The triangular element described by Equation 1.1 is said to have 3
degrees
of freedom,
as three nodal values of the field variable are required to describe
the field variable everywhere in the element. This would be the case if the field
variable represents a scalar field, such as temperature in a heat transfer problem
(Chapter 7). If the domain of Figure 1.1 represents a thin, solid body subjected to
plane stress (Chapter 9), the field variable becomes the displacement vector and
the values of two components must be computed at each node. In the latter case,
the three-node triangular element has 6 degrees of freedom. In general, the num-
ber of degrees of freedom associated with a finite element is equal to the product
of the number of nodes and the number of values of the field variable (and pos-
sibly its derivatives) that must be computed at each node.
How does this element-based approach work over the entire domain of in-
terest? As depicted in Figure 1.1c, every element is connected
at its exterior
nodes
to other elements. The finite element equations are formulated such that, at
the nodal connections, the value of the field variable at any connection is the
same for each element connected to the node. Thus, continuity of the field vari-
able at the nodes is ensured. In fact, finite element formulations are such that
continuity of the field variable across interelement boundaries is also ensured.
This feature avoids the physically unacceptable possibility of gaps or voids oc-
curring in the domain. In structural problems, such gaps would represent physi-
cal separation of the material. In heat transfer, a “gap” would manifest itself in
the form of different temperatures at the same physical point.
Although continuity of the field variable from element to element is inherent
to the finite element formulation, interelement continuity of gradients (i.e., de-
rivatives) of the field variable does not generally exist. This is a critical observa-
tion. In most cases, such derivatives are of more interest than are field variable
values. For example, in structural problems, the field variable is displacement but
(
x
,
y
)
=
N
1
(
x
,
y
)
1
+
N
2
(
x
,
y
)
2
+
N
3
(
x
,
y
)
