Ex-6: Linear Elements vs Quadratic Elements

First, it’s important to understand exactly what differentiates linear elements from quadratic elements. The difference between basis order 1 (linear elements) and basis order 2 (quadratic elements) is how closely the elements conform to the geometry. This idea can be seen in here:

This means fewer quadratic elements than linear elements are required to accurately represent the geometry in simulation. Quadratic elements are more computationally expensive, so start with a low resolution than if linear elements were used.

Generally speaking, this depends on the dominant deformation “type”. If the geometry or a portion of its geometry is under strong bending, quadratic elements generally perform better. Typically we see this in thin (slender) beam and plate/shell geometries with high aspect ratio (where beam cross-section and plate/shell thickness are two or more orders smaller than their span).

To demonstrate this, the following simulation was set up. The out of plane shear in the z direction was used to measure the differences in performance of the linear elements and quadratic elements.

The beam has a cross section of 0.2 X 0.1 m2 (yz plane) and a length of 6.0 m (x-axis).

One end of the beam is fixed, while a unit load is applied to the other end in x,y, and z directions, respectively, as well as a unit twisting load around the x axis.

To be clear, both of these methods work and will return accurate results. Changing the basis order can optimize the amount of time it takes to return an accurate result. Generally <1% error is considered good, so we compared the amount of time linear elements and quadratic elements converged to an error of <1%.

The results can be seen here:

Notably, the resolution for the linear element test needed to be much higher to achieve an error of <1%. Even with fewer elements, the quadratic elements had a smaller error.

For thin geometries, linear elements fail at high aspect ratios. This is because as the elements decrease in size to accurately capture the geometry, the total number of elements increases drastically. This relationship can be seen below. Quadratic elements can capture the geometry at a cell size larger than the thinnest portion of geometry. This makes them especially good for geometries that include a thin portion.

Choosing an element type will differ with each geometry. We recommend using quadratic for bending scenarios first, but if the geometry is very complex, linear elements may perform better.

Footnote: Figure 1 Source: femor_lecture_2.pdf