One efficient method of utilizing finite element (FE) modeling is to exploit the planes of symmetry or anti-symmetry in an assembly or a part being analyzed. When we can take advantage of symmetry, we need to model only a portion of the actual structure and this reduces the analysis run time and memory required. The lines or planes of symmetry or anti-symmetry in an FE model can be simulated by providing proper restraints to the symmetrical faces or edges. To model symmetry or anti-symmetry, the geometry and the restraints must be symmetric about a plane. The loads must be either symmetric or anti-symmetric. Both symmetry and anti-symmetry restraints are discussed in detail in the following sections.

Symmetry/anti-symmetry restraints

For solid models using solid finite elements, the symmetry or anti-symmetry restraints are restricted to only translational displacements. This is because the solid models are modeled only with either TETRA4 (Low) or TETRA10 (High) quality elements in COSMOS™, which do not have rotational degrees of freedom. Consider a plate simply supported at both ends subjected to symmetric loading as shown in Figure 1.

Figure 1: Solid model with symmetry plane

For this model the symmetry restraints on the symmetry plane are Uz = 0; Uy = free, and the anti-symmetry restraints are Uz = free; Uy = 0. (Note that the z direction is normal to the cut face). The displacement response of a full plate due to symmetry and anti-symmetry loading are shown in Figure 2 below. It can be seen that at the plane of symmetry, the applied displacement restraints are not violated. The identical results for a half symmetry model would be obtained using the same restraint conditions at the symmetry plane.

Figure 2: Symmetric and antisymmetric deformation

For shell elements, in addition to the translational restraint, a rotational restraint also needs to be applied along the edge of the symmetry. Figure 3 shows the shell model and the associated symmetry and anti-symmetry restraints.

Figure 3: Shell model with symmetry plane

Frequency and buckling response

In the previous discussion, the tacit assumption was that under symmetric loads, a symmetric structure with symmetric restraints will respond symmetrically, and with anti-symmetric loads, will respond anti-symmetrically. For buckling or frequency response of a symmetric structure, this is not always the case. Care should be taken when using the symmetry/anti-symmetry restraints for frequency and buckling analysis. Symmetry restraints for a half or quarter model yield only symmetry modes and anti-symmetry restraints will give only the anti-symmetry modes. So on the symmetry plane or edge, the analysis needs to be repeated for every possible combination of symmetry and anti-symmetry restraints. For example, if you were asked to find all the natural modes and frequencies for a symmetric plate and you use a quarter model, then you need to run the frequency study for this quarter plate four times with the following set of restraints on the symmetry planes.

Four runs of frequency study need to be run for each of the four restraint cases on the A & B symmetry planes as shown above. The frequencies obtained from each run put together will form a complete set. For buckling, the lowest (positive) buckling factor obtained from all four studies will define the buckling load. This procedure can be applied to any type of restraint and geometry as long as the symmetry plane or edge can be identified.

Conclusion

By taking advantage of symmetry or anti-symmetry, you can reduce the size of the analysis domain, at least by a factor of two. The reduction in analysis domain could mean the difference between keeping the analysis in-core, or swapping during the analysis. Or, a finer mesh can be used in the reduced analysis domain, resulting in a more accurate analysis than a coarsely meshed full model with a comparable node and element count. This is where the real payoff is for taking advantage of symmetry conditions.