When element results are provided to Patran at quadrature points, it is necessary to extrapolate the results from the quadrature points to the nodes of the element and to the element centroid. Similarly, when results are provided at the element nodes or the centroid, it is necessary to interpolate/extrapolate the results to the centroid or nodes respectively.

Patran has three basic methods to perform this interpolation/extrapolation:

The User Interface allows for four basic methods in which the user can control extrapolation methods. These are explained below and examples given.

If the arrangement of node/quadrature points corresponds to an element type in Patran, the shape functions are known, and a parametric mapping is used. This is the preferred method, and is the most accurate representation. The parametric mapping method involves mapping the output positions to an element topology that interpolation functions of that topology can be used to compute results at the nodes. As an example, if there are 27 results output at 27 quadrature points inside a hex/20, then these 27 quadrature points can be considered as 27 vertices of a hex/27 element. Results at hex/20 nodes are then computed by the interpolation function of the hex/27, even though these nodes are located outside the element formed by the 27 quadrature points. Once the nodal results of the hex/27 are available, results at the nodes of the hex/20 can be computed by interpolation. These results will be stored in a 20x27 matrix of coefficients. This method only works if there exists an element topology in the library that coincides with the output pattern after being parametrically mapped.

If the arrangement does not correspond to a Patran element type, a system of equations is constructed and solved for the unknown nodal and centroidal values. The equations are set up such that if the interpolation functions of the element topology are used with the unknown nodal values, they will generate a unit value at each quadrature point. This method only works if there exists an element topology in the library that has the same number of nodes as the number of quadrature points. If Shape Function is set in the User Interface ,the shape functions or a set of equations will be used to extrapolate results as explained above. Only if these two methods fail will averaging take place.

If both previous methods fail, results in the element are averaged and each node of the element will assume this averaged value. Or, alternatively, if the results are provided at nodes, the nodal values would be averaged and assigned to the centroid.

Averaging is also used at element boundaries. In these cases, when extrapolation from different elements yields different result values at the same node, the different results are averaged and assigned to the node.

For degenerate elements, the extrapolation is performed on the parent element topology, and the results at the duplicated nodes in the degenerate element are then averaged.

The User Interface allows for a forced average extrapolation method to be used. The following scenarios can exist.

A forced extrapolation of the analysis results to the element's centroid can also be set in the User Interface which will be performed relative to where the results are initially located. Shown below are several different cases that can occur. Once each centroid value is established it is then used to render the results plot.

The Min/Max method searches each element's results and finds the minimum/maximum value contained within the element. The element then assumes a constant value (including its nodes). For example if the analysis result values are know at the elements Gauss points the minimum/maximum value is used as a constant value across the element. This method has no effect for results that already exist at the element centroid or the nodes.

Examples are given below for each extrapolation technique using a simple 4 node QUAD element with four interior Gauss points. The Gauss points are located in parametric space at +/- 0.5773502692 (as per theory). In p/q parametric space, where the extrapolation occurs, would look something like Figure 13‑3.

The element will have a simple set of linear shape functions described by

Using these shape functions, the results at any point in the element would be found as

where i runs from 1 to 4 for the four Gauss or grid points.

Note that the shape functions vary by element type and element order. The function shown in these examples are not necessarily the functions used in any particular element formulation; they are to illustrate the extrapolation methods only.

Gauss point results are as follows:

The stress values at the Gauss points will be extrapolated to the grid locations. To do this, the Gauss points are assigned parametric locations of 1.0. The location of the grids will be at parametric locations of 1/0.5774 or about +/-1.7319 with respect to the Gauss points.

The stress at grid 14, located in parametric space at x/y coordinates of (1.7319, 1.7319) will be calculated as:

The stresses at the rest of the grids would be as follows:

The stress at the Gauss points are the same as Example 1. The element centroid would be located in parametric space at (0,0), so interpolation to that point can be accomplished directly:

In this example the results at the grid points are provided to Patran. To make an element fill plot, the element centroidal value must be known. The stress values at the element grid points are:

The value at the centroid is then calculated using the shape functions, just as in Example 2 above:

Note that this gives the same results as in the previous example.

The averaging technique simply computes the mathematical average of the nodal stresses and reports this as the centroidal value. So, the centroidal stress would be reported as:

In this case no suitable set of shape functions exists to carry out a proper interpolation. Therefore, the Gauss point stresses are averaged, and the average result distributed to all the grid points:

The grid point stresses would be reported as:

In this case there is only one piece of stress data available, so no assumptions about the stress distribution can be made. Therefore, if the element centroid stress is reported as 15.00, the grid point stress will be reported as:

In this case the stresses in an adjacent element are included in the reporting of the grid point stress. If two elements have nodal stresses calculated from Gauss points by internal extrapolation as follows:

The nodal stresses calculated by Patran would be: