Several examples of a Lagrange interpolation are discussed at this section. At first the plane triangular domain is considered. The location of the interpolation points are shown at Fig. 7.5.

The location of the interpolation points is sumarized at the table below.

The base vector ; contains the monomial
expressions of a complete first order polynomial as it has been described by
Eq. (7.6).

These interpolation points ;
according to Eq. (7.16) are inserted into the base vector according
to Eq. (7.17) to form the quadratic Vandermonde matrix
; .

The associated inverse matrix ; is shown below.

The three shape functions are finally obtained from Eq. (7.9).

The interpolation points ;
,
according to Eq. (7.16), may be inserted into the shape functions to
check the conditions according to Eq. (7.13), Eq. (7.14)
and Eq. (7.15).

The shape function
of the linear Lagrange
interpolation is sketched in Fig. 7.6. Similar figures of the
shape functions are obtained at the corner points 1 and 2 of the triangular
interpolation area according to Fig. 7.5.

In the next example a Lagrange interpolation is considered with four
collocation points in a square two-dimensional domain. Their location is shown
at Fig. 7.7. The location of the interpolation points is sumarized at
the table below.

The base vector ; contains the monomial
expressions of a complete first order polynomial as it has been described by
Eq. (7.6). In addition the binomial term is provided.

These interpolation points ;
according to Eq. (7.18) are inserted into the base vector according
to Eq. (7.19) to form the quadratic Vandermonde matrix
; .

The corresponding inverse matrix ; is
shown below.

The four shape functions are finally obtained from Eq. (7.9).

The interpolation points ;
,
according to Eq. (7.18), may be inserted into the shape functions to
check the conditions according to Eq. (7.13), Eq. (7.14)
and Eq. (7.15). The shape functions
;
are graphically
displayed over the squared plane domain at Fig. 7.8.

Now a parabolic Lagrange interpolation over the triangular domain is considered. The location of the interpolation points are shown at Fig. 7.9.

The base vector ; contains the monomial
expressions of a complete second order polynomial as it has been described by
Eq. (7.6).

The location of the interpolation points is sumerized at the table below and
shown at Fig. 7.9.

These interpolation points ;
are inserted into the base vector according to Eq. (7.20) to form the
quadratic Vandermonde matrix ; .

The associated inverse matrix ; is shown below.

The six shape functions are finally obtained from Eq. (7.9).

The interpolation points ;
,
according to Eq. (7.21), may be inserted into the shape functions to
check the conditions according to Eq. (7.13), Eq. (7.14)
and Eq. (7.15). With respect to Eq. (7.21) the following
expressions are obtained.

The shape function of the parabolic Lagrange interpolation according to Eq. (7.22) is sketched in Fig. 7.10. Similar figures of the shape functions are obtained at the corner points 2 and 3 of the triangular interpolation area according to Fig. 7.9.

The shape function
of the parabolic Lagrange
interpolation according to Eq. (7.23) is sketched in Fig. 7.11. Similar figures of the shape functions are obtained at the
midside points 4 and 5 of the triangular interpolation area according to Fig. 7.9.

Finally a three-dimensional Lagrange interpolation over the tetrahedral domain is considered. The location of the interpolation points is sumarized at the table below and shown at Fig. 7.12.

The base vector ; contains the monomial
expressions of a complete three-dimensional first order polynomial as it has
been described by Eq. (7.7).

These interpolation points ;
according to Eq. (7.24) are inserted into the base vector according
to Eq. (7.25) to form the quadratic Vandermonde matrix
; .

The corresponding inverse matrix ; is shown below.

The four shape functions are finally obtained from Eq. (7.9).

The interpolation points ;
,
according to Eq. (7.24), may be inserted into the shape functions to
check the conditions according to Eq. (7.13), Eq. (7.14)
and Eq. (7.15).

The graphically representation of these shape functions would require a
four-dimensional space and is threrfore not shown here.

Prof. Dr.-Ing. D. Maurer, Hochschule Landshut