FEM: Mesh quality and body simplification (Part 2) - Symmetry

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      Have ever you asked: What is symmetry in FEM? What are benefits using symmetry in FEM? How to performing symmetry in FEM? In this article, we will briefly answer these questions

      Symmetry:

There are some cases which the problem size can be reduced if a symmetry can be checked. The advantage of that is to decrease the computational time to solve the problem. Another advantage is the possibility to visualize internal stress distribution in the body.  

Figure 1 - Internal stress distribution can be visualized.

To ensure the symmetry of problem, all these conditions must be observed as for symmetry [Erdogan Madenci • Ibrahim Guven]:

•           Geometry;
•           Material properties;
•           Loading;
•           Degree of freedom (DOF) constraints;

It is important to remember that in modal analysis, generally is not possible to use symmetry, as the most of mode shapes are not symmetrical.

Basically, there are four types of symmetry:

Axisymmetry: This symmetry is about a central axis, where the body is created from a revolve shape.

Figure 2 - Example of axisymmetric body and its cross section. FONT: ANSYS

This type of symmetry can provide the most simplified model. For instance, the 3D model above, could be converted to a 2D model using axisymmetry.

Figure 3 - A 3D mesh reduced to 2D using axisymmetry. FONT: [Erdogan Madenci • Ibrahim Guven]

Rotational symmetry: When repeated segments are arranged about a central axis. The body can be created from only one segment, for instance, turbine rotors.

Figure 3 - Example of rotational symmetry. Due to holes, this body cannot be represented as axisymmetry. FONT: ANSYS

Planar or reflective symmetry: When one half of body is a mirror image of the other half. 

Figure 4 - Example of planar or reflective symmetry.

Repetitive or translational symmetry: It happens when segments are repeated along a straight line. In this case, only one segment can be analyzed.

Figure 5 - Example of repetitive or translational symmetry. FONT: ANSYS

      

       Considerations about material proprieties, loadings and displacements:

                We don´t need necessarily have a homogeneous material or loading on the body to have a problem symmetry. In some cases, is just a matter to find where is the symmetry axis or plane to do the problem reduction. Below, some examples of these cases:

Figure 6 - Example of symmetry in non-homogeneous material or loading. FONT: [Erdogan Madenci • Ibrahim Guven]

Figure 7 - Example of a non-symmetry case. FONT: [Erdogan Madenci • Ibrahim Guven]

                Another point of attention of reduction through symmetry is to correction of boundary conditions (loadings, supports, and connections).  For instance, when there is punctual load, in some cases, this load must be reduced proportionally of symmetry reduction (a half, a quarter etc) if the load are on the symmetry plane or axis.

Figure 8 - Due to symmetry condition, the load of right side case must be the half of the left case.

                
                Some adjustments must be done to the constrains too. The faces or edges of the reduced body, must have the same behavior as the complete body. It means to create extra supports that mimic the natural behavior of body without its reduction.

Figure 9 - Extra supports must be added to mimics the natural behavior of complete body. FONT: ANSYS

                There are some considerations used to solid, surfaces or bars/beams model. For instance, in a solid model, where there are only translational freedom degrees, the constraint only must be translational of the perpendicular direction of symmetry plane. In the surface model, with six freedom degrees (ux, uy, uz, rx, ry and rz), the constraint must be translational of the perpendicular direction of symmetry plane and for two rotations of the two axes that former the plane. In a beam with two translational and one rotational freedom degrees, the constraint must be translational of the perpendicular direction of symmetry plane and to rotation of the perpendicular axis of the model plane. This reasoning can be used for others similar problems.

Figure 10 - Translational constraint “uy” to solid model. One translational “u” and two rotational “r” for each symmetry edge of the surface model. One translational “ux” and rotational “rz” constrain for beam model.