# Cyclic Symmetry Analysis

Cyclic symmetry is a type of symmetry in which a representative (or basic) segment, if patterned circularly about an axis of symmetry would result in the full model.

Structures with symmetry can often be modeled with one representative entity, from which the full structure can be obtained after certain operations (pattern, mirror, rotation, etc.). Such methods can lead to huge savings in modeling effort, computational time, and file storage. Therefore, exploiting symmetry in finite element models could prove to be very advantageous.

Aircraft engine turbines, gas turbine compressor wheels, windmill assembly, vehicle rims, flange joints are some common examples of structures with cyclic symmetry. The images (Figure 1) show the axis of cyclic symmetry and the representative or basic segment.
Note:
1. The basic segment could either be the smallest repeating unit in the model or could involve several repeating units.
2. Depending on the user’s requirements, a full 360 degree model or a partial (as in, 270 degrees) model could be analyzed by suitably specifying the number of segments in the model definition.

## In OptiStruct

In a cyclic symmetry analysis, the boundaries on either side of the modeled base segment which are to be connected to adjacent segments are defined as SIDE 1 and 2. The direction of the axis of symmetry are determined using the right-hand thumb rule.

If the fingers are curled from SIDE 1 to SIDE 2, then the direction of the thumb determines the direction of the axis of symmetry.
The 1st segment corresponds to the one which is modeled (base segment). Using the right-hand thumb rule, if the right-hand thumb is along the positive direction of the axis, then the direction of along which the fingers are curled determines the direction along with subsequent segments are numbered (Figure 3).

In a cyclic symmetry analysis, the response of the full model is determined from a linear combination of several independent basic responses, known as solution harmonics. They are represented using indices called harmonic indices and they could be selected by the user during the modeling process.

The harmonic indices are supposed to be non-negative and must be no greater than (NSEG is the number of segments to be modeled).

If NSEG is odd:(1)
$\left(\frac{NSEG-1}{2}\right)$
If NSEG is even:(2)
$\left(\frac{NSEG}{2}\right)$
In general, lower harmonics lead to the highest contribution. Depending on the analysis type and loading, solution harmonics play a suitable role, as summarized.
• It is recommended to use all the solution harmonics, to obtain same result as the full model.
• Removing some harmonics could lead to loss in accuracy.
• For large models, removing higher harmonics could help with reduction in the CPU time.
Linear Static Analysis with Non-cyclic-symmetric loading In this case, only the 0th Harmonic will be used, and all other harmonics will be disregarded.
Normal Modes Analysis Removing some harmonics will only lead to some missing mode shapes, without any loss in accuracy.
The following tables summarize the relevant input file entries in a cyclic symmetry analysis.
Table 1. I/O Options and Subcase Information Entries
HARMONICS Specifies the solution harmonics to be used.
• This entry is optional for cyclic symmetry analysis.
• By default, all the solution harmonics are used.
NOUTPUT Specifies the segments for which results must be recovered and output.
• This entry is mandatory for cyclic symmetry analysis.
• If a segment is not specified, the results are not recovered and output. Therefore, such segments will not be shown in the result plot.
Table 2. Bulk Data Entries
CYAX Specifies the grids that lie on the axis of symmetry. This entry is optional for cyclic symmetry analysis.
CYJOIN Specifies grid points on the segment boundaries that connect to adjacent segments.
• This entry is mandatory for cyclic symmetry analysis.
• Two entries, one for each side of the segment will be required.
• Each grid on a CYJOIN entry should be paired with a matching grid on the other entry.
• The axis of cyclic symmetry is determined by the geometry and CD fields of the first pair of grids appearing on CYJOIN entries.
CYSYM Specifies the number of segments in the model. This entry is mandatory for cyclic symmetry analysis.
• This entry is required when referencing loads in a linear static subcase, for cyclic symmetry analysis.
• This is not required, if LOADCYH is used to specify the loads.
• This entry is required when referencing loads in a linear static subcase, for cyclic symmetry analysis.
• This is not required, if LOADCYH is used to specify the loads.

## Support Information

The current support of cyclic symmetry analysis in OptiStruct is:
Table 3. Support table for cyclic symmetry analysis
Analysis type Linear static analysis

Normal modes analysis

In normal modes analysis, only Lanczos eigensolver (EIGRL) is supported.
Analysis output DISPLACEMENT

STRESS

STRAIN

• Only .h3d output format is available.
• Beam element stress/strain is not available.
Elements CBEAM

CTRIA3, CTRIA6

CTETRA, CHEXA, CPENTA

Constraints MPC

RBE2

RBE3

RBE2 and RBE3 will be generated only for the base segment in the result plot.
Materials MAT1

MAT2

MAT9

MATT1

MATT2

MATT9

Loads Enforced displacement (only via non-zero SPC)

Forces

Pressures

GRAV

RFORCE

TEMP

• GRAV is only supported with rectangular coordinate system.
• RFORCE is only supported for rotation about the axis of symmetry.
• Subcase must be linear static.

## Problem Setup

Below is an example of a typical cyclic symmetry analysis setup in an input file.
$*************************************************************$ EXAMPLE TO DEMONSTRATE A CYCLIC SYMMETRIC ANALYSIS SETUP
$************************************************************* . . NOUTPUT = ALL SUBCASE 101 SPC = 1 LOAD = 14 SUBCASE 102 SPC = 1 METHOD(STRUCTURE) = 2 HARMONICS = 102 BEGIN BULK$--1---><---2--><---3--><---4--><--5---><--6---><---7--><--8---><---9-->
CORD2C        11       0     0.0     0.0     0.0     0.0     0.0     1.0
+            1.0     0.0     0.0

EIGRL          2     0.0              50                             MAX

SET          102    MODE
+              0       1
CYSYM          4
CYAX         100
CYJOIN         1              12    THRU      20
CYJOIN         2              22    THRU      30

FORCE         51      12      11     1.0     0.0   100.0     1.0
.