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The Lost Art of Anti-Symmetry

FEA Model | FEA Consulting
April 26, 2017 By: Peter Barrett

When I first learned how to perform finite element analyses in the early 80’s, nearly all the models created used some kind of symmetry.  This was a necessity due to the lack of computing power at the time.  Most of the analyses were 2D and many used a combination of symmetric and antisymmetric boundary conditions to simulate the unique loading environments. In previous posts I have discussed 2D analyses and axisymmetric simulations as ways of simplifying finite element models. In this post, I will discuss the lost art of anti-symmetry, which, along with superposition and symmetry, can be used to solve asymmetric loadings with significantly reduced computational resources.

Figure 1, above, illustrates a plate model that can be solved with simplified geometry along with a combination of symmetry and anti-symmetry boundary conditions. What is the smallest possible model and corresponding supports that can be used to solve this problem?  The answer will be provided at the end of this blog (no cheating - don't read ahead!)  Before we discuss anti-symmetric boundary conditions, a refresher on symmetry is provided.

Symmetric boundary conditions can be used in 2D and 3D models constructed from beams, shells and solids. The premise for modeling symmetry is that the geometry, material and loading conditions are identical on either side of the reflection plane. Symmetry boundary conditions inherently provide partial restraints against rigid body deformation and do not introduce any artificial singularity, as can often occur with fixed support boundary conditions.

Consider a simple beam model with two point force loads as illustrated in Figure 2. First the full beam is simulated and then ½ the beam is analyzed, but with the addition of symmetry at the centerline.  In order to enforce symmetry, displacement constraints are imposed on the reflective boundary. For 3D and 2D solid element meshed models, only displacements normal to the symmetry plane need to be constrained. Beam and shell elements require additional restraints to fix the two in-plane rotations. Figures 2 through 4 illustrate a full vs. ½ symmetric beam modeled with solid, shell and beam elements.

 

Figure 2 - Full vs. Symmetric Solid Model Example

 


 

Figure 3 - Full vs. Symmetric Shell Model Example

 




Figure 4 - Full vs. Symmetric Beam Model Example


Antisymmetric boundary conditions are defined in a similar manner as symmetry, where a mixture of displacement and rotation constraints are applied on the symmetry plane of the model.  With antisymmetric boundary conditions, the opposite degrees of freedom from the symmetry degrees of freedom are constrained. For solid elements, the two in-plane displacements are constrained, while for beams and shells an added rotational constraint is applied about the axis normal to the symmetry plane. Hence, when comparing the symmetric and anti-symmetric boundaries, the combination of both symmetric and anti-symmetric conditions results in full fixity of all active degrees of freedom on the symmetry plane. (This was an important fact to know when using version 4 of ANSYS in the late 1980’s since, the easiest way to completely fix a surface was applying both supports as cumulative boundary conditions.) Figure 5 illustrates the same response with the full vs. antisymmetric loading on the sample beam element model. One limitation that should be noted is that, while a symmetry boundary condition is applicable for both small and large deflection simulations, the antisymmetric assumption is limited to small deformations.  If the displacements or rotations at the symmetry cut are expected to be large, the antisymmetric assumption should not be used.


Figure 5 - Full vs. Anti-Symmetric Beam Model Example

 

 

Strictly antisymmetric loads by themselves are not that common, but anti-symmetric models can also be used to solve asymmetric loading. For example, if the model requires loading that only occurs on one side of the symmetry plane then neither the symmetric or antisymmetric loads by themselves are applicable. However, when one uses superposition, one can replicate the asymmetric response.  The symmetry boundary creates the equivalent of the same load on each side of the symmetry plane, while the antisymmetric load represents an equal but opposite load on the opposite side of the antisymmetric plane. Thus, analyzing 50% of force applied to at the same location to both the symmetric and antisymmetric load cases independently and then adding the results together via postprocessing produces a asymmetric loading response. For the simple beam example, the asymmetric case is simulated using 50% of the force loading solved first with the symmetric boundary condition and again with the antisymmetric boundary condition. A load combination procedure of superposition is then used to simulate the full asymmetric solution. For the loaded side of the beam, the two results sets are viewed by simply adding the two analyses together, while for the unloaded side of the beam the symmetric and antisymmetric solutions are subtracted from one another. Figure 6 illustrates the more interesting combined solution which matches the full asymmetric beam simulation. This method of superposition is only valid for linear problems.


Figure 6 - Asymmetric Full vs. Superposition of 1/2 Beam Model Example


So back to our original question. Figure 7 illustrates how a 1/8 model with a mixture of symmetric and anti-symmetric boundaries can be used to solve the quiz. 



Figure 7 - Smallest Repeatable Segment and Boundary Conditions Required to Solve Quiz Problem


Do you use antisymmetric in any of your FEA simulations?  I would love to hear others’ experiences in this lost art of simulation.