A study of optimal membrane triangles with drilling freedoms

A family of triangular and tetrahedral elements with rotational degrees of freedom

“…Figure 4 contains results using three different triangular element formulations and different mesh resolutions. The curve labeled as QTRI3 is the solution using the element proposed in this [9]. Both solutions are very close to each other.…”

“…Note that the nodal rotations r (k) are introduced as additional degrees of freedom which are to be interpreted as relative rotations superimposed on the displacement field and the average rotation generated by the nodal displacements u (k) and v (k) . This notion of "hierarchical rotations" has been discussed in [1,2] and elaborated on in detail in [9]. Hence the nodal rotations are formally defined as…”

“…Plane stress conditions are assumed. The results from the element QTRI3 as developed in this paper are again compared to those using the optimal element [9] and the constant strain triangle. In this example, it can be observed that the QTRI3 element converges quickly from below whereas the OPT element converges from above.…”

“…Again, the constant strain triangle is overly stiff. It should be noted that the displacement of the point C (0.35601 as given by [9]) is taken as reference (i.e., as 100%). This solution is slightly larger than the analytical solution of 0.35533 as reported in [9].…”

“…In order to circumvent these problems, several strategies have been proposed, such as the assumed strain concept (e.g., [4,10,14]) or the introduction of additional degrees of freedom, e.g., drilling DOF's [1,7,12,15,18]. A very general overview is given, e.g., by [9]. This paper aims at laying out a methodology to derive simplicial element formulations based on nodal translations and nodal rotations.…”