Nodal vs Element Results

Quick demonstration of how results are calculated and presented in FEA as Element or Nodal

This simple model shows how results such as strain and stress are calculated at the centre of an element and then are averaged at nodes by finding the average of all connected elements.   

The model has 2 plane strain elements, the nodes at the bottom are fully constrained and single point load is applied in an upper corner.

As results averaging is set to “None”, each element has only one result value and a single block colour.  The fringe scale runs from max to min, hence one element gets the max colour, the other gets the min colour.

The lower element (1), has ex = 0*.  The upper element (2) has ex = -13579ue.   This is handy for the demonstration.

Nodes 3 & 4 are connected to both elements hence get the average of two element values.  (0 + -13579)/2 = -6790.    Nodes 1 and 4 are only connected to one element so their value will be equal to the relevant element value.

To see nodal results, change Result / Averaging Type to All.  Now the nodal results are presented in text at each node and the fringe colours linearly interpolate between the 3 nodal values for each element.  

The fringe colours naturally match at the boundary of the elements as they share the same nodal values on this edge.

Further Notes

Therein lies the danger however as nodal results can hide the discretisation error.  Sometimes looking at  elemental results gives us a feel for how much discretisation is present.  

ex = 0 for element 1 because it is a plane strain element and when fully constrained in x and y for two of its nodes (1 & 2), the ex value will be zero. 

Quickfem actually uses an area weighted average nodal average where the element areas are made part of the nodal averaging process.  In this example the areas of the elements are the same anyway.

Advanced FE packages use higher order elements where element results are calculated at multiple locations within the element (called Gauss points).  There are then multiple ways of averaging or interpolating the results to the nodes.

Refinement of the model will eventually remove all sins of discretisation and approach the theoretical solution.

Usually, using point loads in any model, especially heavily refined models is not recommended as they create singularities.  Better to use distributed loads.