A3. Choosing the finite elements

Choosing the element is an important step. The goal is to select the type of element (its shape, the order of the shape functions associated), and the size of the element. The type and size of the elements define the form and the precision of the displacement fields, and consequently the stresses and strains.

Moreover, besides the shape, the element has an aspect. It should be avoided to create decayed or altered elements (flattened or elongated) because it degrades the precision in the resolution of the problem.

Generally, the generated mesh can be structured (a regular division of the elements) or unstructured. It is possible to mix structured and unstructured parts in the same domain depending on the complexity of the geometry.

In the same context, the size of the element chosen depends on the geometry of the structure to model and the loadings it is subjected to. The zones with high-stress variations (gradients) or high stresses (friction or cracking, for example) determine the parts where the mesh needs to be more refined in comparison with the other parts to properly observe the stresses and the strains.

It is important to perform the first simulation with an initial meshing to determine the sensitive zones and then refine the mesh where needed.

The person in charge of modeling must analyze the meshing critically considering the geometry of the structure and the important zones to observe.

To choose correctly the finite elements construct the mesh, it is imperative to think about the type of calculation desired:

• the whole geometry is going to be represented. It is a Continuum Mechanics problem, so the elements are of the “solid” type and the problem must be predefined as three-dimensional or two-dimensional (plane stress, plane strain assumptions, axisymmetry… see chapter 3). If one wishes to use linear or quadratic shape functions, the precision of the calculations improves with a mesh of quadratic elements, but with an important addition in the computational cost. Thus, a compromise has to be made. In any case, the degrees of freedom are the displacement components (u, v in 2D, u, v, w 3D). The main elements are listed below,

• the geometry is simplified, in which case one has a Strength of Materials problem (or structural analysis, see Chapter 2) and the finite elements are:
  

• bar/beam elements (a bar element can only transmit tension or compression, whilst the beam element is also capable of transmitting bending moments. Be careful, some software use the term “bar element” to designate any 1D element),
  

• plate/shell elements (the difference between plate and shell is related to the curvature of the mean surface and most software do not make any distinction between the two).
  

Besides the displacement degrees of freedom, the Strength of Material elements have also rotational degrees of freedom (θx, θy, θz), allowing to take into account the non-meshed geometry (cross-section for beam elements, thickness for plates and shells). Moreover, one should ask the question if the shear forces should be taken into account or not (Navier-Bernoulli or Timoshenko beam elements, Love-Kirchoff or Mindlin-Reissner shell elements). In the case of shell elements, as mentioned before, the question about the quality of elements comes to mind (compliant or non-compliant elements). It is particularly difficult to choose between plate and shell finite elements, especially because the literature is scarce in this area. It can be helpful to perform a calculation with known results to test the quality of the available elements.

The different elements that are commonly found in Strength of Materials are described in the table below.