In many problems the value of this will be known in some form; in fact in many cases it will be a constant. However there will be occasions where the explicit value of is not known but is given as the rates of change of with respect to some set of global axes.

For example consider a two-dimensional problem
formulated in a Cartesian coordinate system . The
boundary values of may well be given with respect to this system of
axes. Thus on a boundary

and along DE ()

However, problems present themselves whenever the boundary is not parallel to the coordinate axes, for example along EF. The normal derivative must now be constructed from the known parts. Clearly this will be

where and are the directions cosines of the outward normal at some point along EF, given by

A straight boundary not parallel with the coordinate axes does not present real difficulty since and will be constants. However in the most general case, where the boundary could be any smooth curve, and will be functions of position. Within the context of finite element analysis the direction cosines will be required on sides of elements.

In general the element
in global coordinates is generated from a parent element by
means of some isoparametric transformation. The most
used transformation is

(4.19) | |||

(4.20) | |||

(4.21) |

where the are a set of shape functions (see Chapter 2) and are the coordinates of the element nodes in global coordinates. The are defined in terms of a coordinate system natural to the parent element (most often called local coordinates).

As an example, consider the triangular elements in Figure 4.2. Figure 4.2(a) shows the elements in the global coordinate system and Figure 4.2(b) shows the parent element. The boundary is approximated by a series of element sides; for each side the outward normal is required.

The construction of these direction cosines is based on two
results (Rutherford, 1962).
Firstly any element side can be presented as part of a set
of * equipotential surfaces* in the coordinate system.
These are characterised by

The second is a result from coordinate geometry:

whereIf at any point P on the surface , then is perpendicular to the surface, passing through P,

(4.23) |

and just as this applies to the elements in the global system it will also apply to the parent element in the coordinate system.

In the local coordinate system the direction cosines
of the
outward normal to any of the sides are simple to calculate.
As in the global case these are proportional to the components
of , where is the equation of one of the element sides; hence

and using (4.20) the direction cosines of the outward normal to the parent element can be transformed into the outward normal to an element in the global coordinate system.

Only the direction of the normal is given so finally the vector is normalised. The components of this normalised vector will be the direction cosines and . Within the context of a finite element program (Segment 3.2 is the best example) the code to calculate the direction cosines is contained within the boundary integration loop since the normal direction will be required at the Gauss points on the boundary:

CALL MATMUL(LDER, ILDER, JLDER, GEOM, IGEOM, JGEOM, JAC, * IJAC, JJAC, DIMEN, NODEL, DIMEN, ITEST) CALL MATINV(JAC, IJAC, JJAC, JACIN, IJACIN, JJACIN, DIMEN, * DET, ITEST) CALL DCSTRI(JACIN, IJACIN, JJACIN, SIDNUM, COSIN, ICOSIN, * ITEST)The routine