Stress Results Smoothing Options - “Smooth before applying operators” or not

[PipePakPat]

Abstract

This essay will examine the details of the Results Environment smoothing calculations and discuss best practices towards producing accurate stress results.

 

Figure 1: Smoothed Stress Result

Introduction

The concept of "smoothing" is method of displaying a continuous element based result across adjacent elements.  Smoothing is aesthetic benefit for the viewer of the results rather than providing an accurate reflection of the calculated elemental results.  This visual enhancement has become the expected default results display in finite element analysis results viewing packages because the display without this option does not look as attractive.  For example, if two software packages provided the same results and one package displayed these results using smoothing and the other didn't, then the user would probably prefer (and purchase) the one which provides smoothing because it "appears" to be reflective of a natural stress display.

 

Figure 2: Un-Smoothed Stress Result

What is result smoothing?

Finite element analysis produces two types of results; nodal based results (such as displacements) and element-nodal based results (such as stress).  For a stress analysis, the definition of the original problem states that the displacements at specific nodes are unique.  This means that for any node between elements, there is only one result.  This is not true for element nodal results such as stress.

 

Figure 3: Four Elements Share one Global Node

Each element has its own nodes that describe its geometry.  The figure above illustrates four elements (in red) and their relation to the common shared global node (black circle).  The element nodal result appears as a red circle at the corner closest to the black circular global node.  The stress field is calculated based on the solved displacements at the element nodes for each element.  The calculated stress field is not continuous from one element to another.  If the smoothing option is not enabled, then there is an obvious difference in stress from one element to the adjacent element.  "Smoothing" is a method of averaging the element nodal results at that global nodal location and displaying a unified result at that location.  But, there are more than one way to producing this unified result.

The two methods are reflected by the "Smooth before operators" option.  Here "Smooth" means to average and "operator" means to determine the calculated stress.  In this case, the calculated stress is the von Mises stress.  There are other calculated stresses, such as Tresca*2.  The results environment reads the analysis results file, which contains element results.  Each element record contains stress tensors for each element node.  From these core results, the von Mises stress must be calculated and displayed in the Results Environment.  The smoothed result can be calculated in two ways.

 

Figure 4: Smoothed von Mises Calculation

“Smooth before applying operators” is Un-checked

From the example above, we can follow the purple dotted line to the final smoothed von Mises stress.  From each element node's stress tensors, the von Mises stress can be calculated, then for each global node, all element nodal von Mises stress results are averaged and that "smoothed" results is displayed at each global nodal location.

“Smooth before applying operators” is checked

The second way would follow the light blue path above to display the smoothed result.  First, for all element nodes that share a global node, the average each of the stress tensors is calculated.  Then calculated the von Mises stress based on those averaged stress tensors.

From the fictitious example shown above, we see there is a large difference in the final calculated von Mises stress and that the value produced with the “smooth before applying operators” option activated was much lower than the value calculated without this option activated.  This example suggests that by using the “smooth before applying operators” option has the potential to demonstrate lower peak stresses than models which don’t use this smoothing option.

NOTE: It should also be noted that this fictitious example would reflect a non-converged stress value and that perhaps refining the mesh would produce more consistent results between adjacent elements.

Why are there two methods of producing a smoothed result?

Years before Autodesk acquired the Autodesk Simulation Mechanical software, the pre-acquisition company provided the method of smoothing for which the von Mises stress was calculated at each element node, then those results were averaged across element nodes that shared global nodal locations.  During software demonstrations, prospects would provide test models so that we could demonstrate consistency with competitive software package results.  Sometimes there were differences in stress between compared results and this inconsistency needed to be explained to build confidence in the calculated result.  At that time, the "Smooth before operators" option was added so that we could then produce the same smoothed results as were produced by competitive software packages.

Can accurate continuous elemental results be obtained?

Smoothing is a shortcut to producing a continuous stress display across a model.  But there is a different and more accurate way to accomplish this same goal without using smoothing.

 

Figure 5: Stress Results Extrapolated to Corner Nodes

One reason that element based results are not continuous between shared global nodes, is because element nodal results are extrapolated values based on calculations performed at Gaussian integration points that lie inside the bounds of the element.  This means that the extrapolated element nodal stresses will either be larger or smaller than the actual calculated values found at the Gaussian integration points.  Also, larger elements can obscure the details of stress over that volume.  The solution towards resolving both of these approximations is to create a model with smaller elements.  As the elements become smaller, then the stress difference between adjacent element nodes also becomes smaller.  Finally, the difference between the smoothed and unsmoothed results approaches zero.  The measurement of this difference is the Precision of von Mises Stress.

 

Figure 6: Precision of von Mises Stress

Note:  The precision of von Mises stress should be observed in the region of the model where significant stresses are produced.  The maximum precision displayed in the legend box may in fact describe a significant stress change in a portion of the model which demonstrates insignificant stress values.

Finite element analysis is a balance between obtaining accurate results within an allowed amount of time and effort.  Also, there are computer limitations as well which limit the size of the finite element model and the time it takes to perform the analysis.  To create a model's mesh, analyze and view results is only the beginning.  The responsible approach would require no less than three analysis iterations,  starting with a coarse mesh and ending with a fine mesh so that a mesh convergence stress plot can be created for a specific point in the model.  A converged stress result approaches an asymptote.  It is unlikely that we reach the exact solution, but can reach a converged solution to within an acceptable tolerance of results change between different mesh sizes.  A converged mesh size may show little difference between adjacent element nodal stress results.

 

Figure 7:  Graph of Mesh Convergence

Conclusion

The most correct result comes from the unsmoothed display.  If the mesh is fine enough, then smoothing will not change the display significantly.

Of the two smoothing options it is more accurate to preserve the stress tensor signs before attempting to calculate the von Mises stress rather than averaging these stress tensors first before performing the von Mises calculation.  I find that the "Smooth before operators" produces  lower calculated stress results than without this option active.

[tibor121774]

Nice presentation Pat....Gives me an idea of proper interpretation of results... Actually I've got a sample simulation...One modelled in Plate Element with Von Mises "Smoothed" results of 87 Mpa, and unsmoothed Von Mises result is 108 Mpa...I've modelled the same using solid element with 140MPa as smoothed Von Mises Result and 351MPa as unsmoothed result...A large difference there in solid element...Maybe I need to investigate on aspect ratios and refine the mesh...Cheers!!!!

[sarajampen12]

Thank you for sharing this information and your experience!