Manage SOLIDWORKS Simulation Mesh Refinement using an Energy Norm Error Plot

Article by Mehdi Rezaei, CSWE updated April 28, 2018


When it comes to computation with SOLIDWORKS Simulation finite element analyses, we are always looking at integrals. Evaluating those integrals need numerical methods. Gauss-Legendre numerical integration is commonly used in finite element codes for this purpose. The reason is this technique shows a higher ratio of accuracy on computing side.

Stress results in SOLIDWORKS Simulation are first calculated at certain locations, called Gauss or quadrature points, which is located inside each element. The following image shows Gauss points for a few draft quality 1st order tetrahedral elements versus their nodes:

Gauss Points vs. Element Nodes

Gauss Points vs. Element Nodes

Note: First order tetrahedral elements (draft quality) have one Gauss point in their volume. Second order tetrahedral elements have four Gauss points. First order shell elements have one Gauss point. Second order shell elements have three Gauss points.

The solution results on Gauss Points results in element stress values meaning the end result has one color per element and we can’t see the nice continuous color transition through elements. To see continuous color transition, FEA codes provide nodal solution where stress values are calculated at the nodes of each element by extrapolating the results of Gauss points. In the following images, elemental solution stress plot is compared to nodal solution stress plot. To make the elemental solution plot closer to the nodal solution plot, element size needs to become smaller. However, there is a limitation which ends up to a trade-off between the element size and computation time. This is a common concern for all finite element simulations.

Stress Plot Using Element Values Option Selected

Energy Norm Error Plot

Stress Plot Using Node Values Option Selected

Therefore as the real calculation happens on Gauss points and then we see extrapolated values on element nodes, this estimation method results in some errors. Energy Norm Error plot shows the error between elemental and nodal solutions results. The smaller the element size, the closer the distance between nodes and Gauss point of an element, the less error and consequently, the more accurate the results will become. However, as we can’t refine the mesh at all model due to long computational time issues, we can use the Energy Norm Error plot to help us to refine the crucial locations and come up with a trade-off between the error value and the element size. The Energy Norm Error plot is calculated based on strain energy principles. Simulation estimates the energy norm error for every element. Here is the formula software follows based on SOLIDWORKS Help:

Energy norm error (element) = 1 /3 * strainerror * stresserror * (Element volume)

The strainerror is the difference between the nodal strain and element strain on an element basis. The stresserror is the difference between the nodal stress and element stress on an element basis.

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Mehdi Rezaei, CSWE

Mehdi is a Certified SOLIDWORKS Expert (CSWE) and works near Vancouver, British Columbia, Canada