Symmetry as it relates to viewfactors and thermal radiation analysis is not a very easy topic to understand. If the user does not plan to make use of symmetry in modeling the problem, then this section may be omitted from study. Also, the user may wish to come back to this section at a later time, after mastering other aspects of viewfactor analysis, since this material is not in the mainstream of the rest of this document. The symmetry which we refer to here is not the symmetry of axisymmetry or the symmetry which allows us to reduce a three-dimensional model to a two-dimensional model. Rather it is the symmetry which pertains to reflections about lines or planes and discrete rotations about an axis.

When modeling thermal problems the analyst may look for and make use of symmetry in the model geometry, materials, and boundary conditions. This is typically done to reduce the overall size of the computer descriptions of the problem and reduce the time required to analyze it. There are many existing models for particular thermal problems and many of them have made use of symmetry in describing the problem. It is advantageous, to the greatest extent reasonably possible, to be able to use these existing models for thermal radiation analysis also.

Unfortunately, this is frequently not possible because the radiative boundary condition imposes a much stricter symmetry requirement on the model than do the other boundary conditions, material properties or geometry. The other boundary conditions typically involve only one surface area at a time, whereas the radiative energy interchange at the boundary involves pairs of surface areas and complicated relationships between the surface’s normals, angles between these normals and the intersurface ray, and the distance between the surfaces. In order for the model to possess true symmetry, all of these attributes must exhibit the same symmetry and this is usually not the case.

It is desirable not to make a full model of an otherwise symmetric model just to deal with the nonsymmetry imposed by the radiative boundary conditions. To this end, Viewfactor provides some capabilities for specifying symmetry for the viewfactor analysis that is only needed for the Viewfactor analysis and not needed for the remainder of the thermal analysis. This will allow the use of existing models, in some cases, with only minor modification (addition of the symmetry information), and allows the thermal network analysis to be performed on the symmetric model while the viewfactor calculations are performed on the complete model.

Patran and Viewfactor cannot check the validity of the user’s symmetry description of the model. When ascertaining the nature of the symmetry present (or not present as the case may be) for thermal radiation interchange and viewfactors, there is ample opportunity for error. These errors are often difficult to detect either by inspection of the model or from detecting erroneous analysis results. We are generally not used to thinking in terms of the symmetry requirements for viewfactors between pairs of surfaces and so this symmetry, or lack thereof, is not intuitively obvious to us as is say geometric symmetry. Frequently we work with problems so complex that we are unable to known whether the analysis is correct or not and thus cannot rely on the detection of erroneous results to detect symmetry errors. Some examples are given later in this section to illustrate the difficulty of correctly modeling symmetry for radiative boundary conditions.

The use of symmetry may also in some cases result in computer numerical errors accumulating in such a way that they do not cancel each other out. Thus it is possible to have significantly larger errors in the viewfactors for a model making use of symmetry than in one which does not use symmetry.

The symmetry supported in Viewfactor will be referred to as symmetry operations, for example discrete rotation and reflection operations. These operations are different depending on the coordinate system in which the model is described. In Viewfactor and Patran Thermal, the user may use any one of three coordinate systems, these being two-dimensional Cartesian coordinates, three-dimensional Cartesian coordinates, and two-dimensional axisymmetric coordinates. The symmetry operations supported by Viewfactor and Patran Thermal for Viewfactor analysis will now be described for each of these coordinate systems.

There are two symmetry operations supported in 2-D XY space (two-dimensional Cartesian coordinate system). These are reflection about a straight line in the coordinate axes plane and some number of rotations by some number of degrees about an axis perpendicular to the coordinate plane.

This is analogous to reflections in a mirror. The line must be straight and in the coordinate system plane. An example is shown in Figure 3‑26.

This symmetry operation causes the entire model to be replicated by rotating it as a unit about an axis. The axis must be perpendicular to the coordinate plane. The number of degrees of rotation must be specified as well as the number of times the rotation is to be repeated. For repeated rotation, the object will be recreated at each step in the series of rotations. An example is shown in Figure 3‑26.

Up to two reflections about lines and one rotation about an axis may be combined in 2-D XY models. Successive symmetry operations operate on the model and its symmetric images present at the beginning of the symmetry operation. An example showing combined symmetry operations is illustrated in Figure 3‑27.

There are two symmetry operations supported in 3-D XYZ space (three-dimensional Cartesian coordinate system). These are reflection about a plane and some number of rotations by some number of degrees about an axis of rotation.

This is analogous to reflections in a mirror. The mirror and the plane representing it must be planar. Its orientation in space does not matter. An example is shown in Figure 3‑28.

This symmetry operation causes the entire model to be replicated by rotating it as a unit about an axis. The number of degrees of rotation must be specified as well as the number of times the rotation is to be repeated. For repeated rotation, the object will be recreated at each step in the series of rotations. An example is shown in Figure 3‑28.

Up to three reflections about planes and one rotation about an axis may be combined in 3-D XYZ models. Successive symmetry operations operate on the model and its symmetric images present at the beginning of the symmetry operation. An example showing combined symmetry operations is illustrated in Figure 3‑29.

There is only one symmetry operation supported in 2-D axisymmetric RZ space. It is reflection across a straight line perpendicular to the Z axis. The straight line must be defined in the RZ plane. No combination of symmetry operations is allowed in this coordinate system. An example is shown in Figure 3‑30.

Figure 3‑31 shows a two-dimensional model of a long solid cylinder surrounded by an annulus and a concentric hollow cylinder. The cylinders are assumed to be long enough that end effects can be neglected and are modeled in two dimensions.

The lower semicircular outer boundary of the outer hollow cylinder receives a nonuniform heat flux as shown by the length of the arrows. Energy is transferred from the inner surface of the hollow cylinder to the solid cylinder by thermal radiation.

Although the geometry is axisymmetric, the heat flux boundary condition is not. The heat flux boundary condition and the geometry together are symmetric about a vertical line through the center of the model. Unfortunately, the radiation boundary conditions are not symmetric about this line as can be seen by examining the view from point 1 to points 2 and 2' (or many other pairs of points).

Normally, this lack of symmetry either would not be noticed by the analyst and an incorrect model created or the entire object in 2-D would be modeled. Viewfactor, through its symmetry operator, will allow the user to model this object as its symmetric right or left half, thus reducing the size of the Patran Thermal analysis by half.

To handle the nonsymmetry of the radiative boundary conditions, the user must tell Viewfactor, through use of the symmetry operators, to take into account the symmetric image of the model when calculating viewfactors and making Patran Thermal radiation resistors. In this case, this would be done by specifying a symmetric reflection about a line coincident with the vertical symmetry line.

Instructions on how to enter the symmetry operators into the Patran model will be given at the end of this section. See Entering Viewfactor Symmetry Operations in the Patran Model, 75.

This example is similar to the previous example except the heat flux boundary condition has been changed to be symmetric on a quarter section of the model. One way to model this problem appears to be to take the upper right quarter section and replicate it by rotating about the center point by 90 degrees and then reflecting the resulting model and image about a horizontal line through the cylinder’s center as shown in Figure 3‑32.

The geometry and boundary conditions seemingly appear as we think they should. You might be tempted to define a Viewfactor rotational symmetry operator and reflection symmetry operator for this problem and proceed with the thermal analysis using the quarter section model. This, however, would be erroneous.

Upon careful examination of the model the reader can see that, for example, the view from point 1 is not the same as the view from point 1''. Thus this is not a correct model of the radiative interchange in this model.

Such errors are subtle and difficult to detect. In general, we recommend that symmetry not be used in thermal radiation models in order to preclude the possibility of such problems.

The symmetry operators for Viewfactor are entered into the model through Patran as special elements. Patran Thermal’s PATQ then translates these special elements into Viewfactor symmetry operators. The symmetry operators, their relationship to the thermal model, and the images of the model they cause to be created in the Viewfactor program are described in The Purpose of Symmetry in Viewfactor, 67 through Symmetry Operations Supported in Viewfactor, 67. This section describes the mechanics of entering the special elements used to describe the symmetry operators in Patran. The meaning and proper use of the symmetry operators is explained in the previous sections.

The symmetry operators and their corresponding special elements are:

The element property data for rotation about an axis contains the number of times the rotation is to be repeated and the angle of rotation in degrees. The element input property form is found under Element Properties looks like

Get the Input Properties form under Element Properties.

The nodes belonging to the radiation symmetry elements have to be declared as type "I" nodes. To do so one must first create OD elements at the location of the nodes belonging to the radiation symmetry elements. When the OD elements are created go to the element properties menu and select dimension: OD, type: Node type and click on the ‘input data’ button and select ‘information node’ in the ‘node type’ data box. Then select the OD elements created as the application region and click on ‘apply’. Nodes of type I (ignore) will not be translated by PATQ into the Patran Thermal nodes, but will only be used by PATQ to generate the Viewfactor symmetry operators.