General vector calculus may be performed along any arbitrary 2-D or 3-D path through a solid element model. Nodal data, element data, and data stored with element output tables (ETABLE command) may be mapped onto the path and operated on as described below.
A path is defined by first establishing path parameters (PATH command) and then defining path points which create the path (PPATH command). The path points may be nodes, or arbitrary points defined by geometry coordinates. A segment is a line connecting two path points. The number of path points used to create a path and the number of divisions used to discretize the path are input (using Npts and the nDiv parameter on the PATH command). The discretized path divisions are interpolated between path points in the currently active coordinate system (CSYS command), or as directly input (on the PPATH command). A typical segment is shown in Figure 17.1: Typical Path Segment as going from points N1 to N2, for the first segment.
The geometry of each point along the path is stored. The geometry consists of the global Cartesian coordinates (output label XG, YG, ZG) and the length from the first path point along the path (output label S). The geometry is available for subsequent operations.
In addition, position (R), unit tangent (T), and unit normal (N) vectors to a path point are available as shown in Figure 17.2: Position and Unit Vectors of a Path. These three vectors are defined in the active Cartesian coordinate system.
The position vector R (stored with PVECT,RADI command) is defined as:
 | (17–8) |
where:
| xn = x coordinate in the active Cartesian system of path point n, etc. |
The unit tangent vector T (stored with PVECT,TANG command) is defined as:
 | (17–9) |
 | (17–10) |
 | (17–11) |
where:
| x, y, z = coordinate of a path point in the active Cartesian system n = 2 to (L-1) |
| L = number of points on the path |
| C = scaling factor so that {T} is a unit vector |
The unit normal vector N (PVECT,NORM command) is defined as:
 | (17–12) |
where:
| x = cross product operator |
|
{N} is not defined if {T} is parallel to {k}.
Having defined the path, the nodal or element data (as requested by Item,Comp on the PDEF command) may be mapped onto the path. For each path point, the selected elements are searched to find an element containing that geometric location. In the lower order finite element example of Figure 17.3: Mapping Data, point No has been found to be contained by the element described by nodes Na, Nb, Nc and Nd. Nodal degree of freedom data is directly available at nodes Na, Nb, Nc and Nd. Element result data may be interpreted either as averaged data over all elements connected to a node (as described in the Nodal Data Computation topic, see POST1 - Derived Nodal Data Processing) or as unaveraged data taken only from the element containing the path interpolation point (using the Avglab option on the PDEF command). When using the material discontinuity option (MAT option on the PMAP command) unaveraged data is mapped automatically.
Caution should be used when defining a path for use with the unaveraged data option. Avoid defining a path (PPATH command) directly along element boundaries since the choice of element for data interpolation may be unpredictable. Path values at nodes use the element from the immediate preceding path point for data interpolation.
The value at the point being studied (i.e., point No) is determined by using the element shape functions together with these nodal values. Principal results data (principal stresses, strains, flux density magnitude, etc.) are mapped onto a path by first interpolating item components to the path and then calculating the principal value from the interpolated components.
Higher order elements include midside nodal (DOF) data for interpolation. Element data at the midside nodes are averaged from corner node values before interpolation.
Once nodal or element data are defined as a path item, its associated path data may be operated on in several ways. Path items may be combined by addition, multiplication, division, or exponentiation (PCALC command). Path items may be differentiated or integrated with respect to any other path item (PCALC command). Differentiation is based on a central difference method without weighting:
 | (17–13) |
 | (17–14) |
 | (17–15) |
where:
| A = values associated with the first labeled path in the operation (LAB1, on the PCALC,DERI command) |
| B = values associated with the second labeled path in the operation (LAB2, on the PCALC,DERI command) |
| n = 2 to (L-1) |
| L = number of points on the path |
| S = scale factor (input as FACT1, on the PCALC,DERI command) |
If the denominator is zero for Equation 17–13 through Equation 17–15, then the derivative is set to zero.
Integration is based on the rectangular rule (see Figure 16.1: Integration Procedure for an illustration):
 | (17–16) |
 | (17–17) |
Path items may also be used in vector dot (PDOT command) or cross (PCROSS command) products. The calculation is the same as the one described in the Vector Dot and Cross Products Topic, above. The only difference is that the results are not transformed to be in the global Cartesian coordinate system.