Cyclic Symmetry Analysis

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:

The basic segment could either be the smallest repeating unit in the model or could involve several repeating units.
Depending on your 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 1^st 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 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)

(\frac{N S E G - 1}{2})

If NSEG is even:(2)

(\frac{N S E G}{2})

In general, lower harmonics lead to the highest contribution. Depending on the analysis type and loading, solution harmonics play a suitable role, as summarized.

Analysis Type	Comments
Linear Static Analysis with Cyclic-symmetric loading	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 0^th 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
Entry	Purpose	Additional Details
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
Entry	Purpose	Additional Details
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.
LOADCYH	Specifies the harmonic coefficients of loading.	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.
LOADCYN	Defines the loading.	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. The desired segment for applying loads can be specified in this entry. This determines the type of loading (cyclic and non-cyclic loading).

Support Information

The current support of cyclic symmetry analysis in OptiStruct is:

Table 3. Support table for cyclic symmetry analysis
Category	Supported Entities	Additional Details
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 CQUAD4, CQUAD8 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. Temperature loading must be referenced in the TEMPERATURE(LOAD) Subcase Entry.
Preloading	Preloaded linear static and normal modes analysis are supported	The following restrictions apply to the subcase that preloads: Subcase must be linear static. Loading in this subcase must be cyclically symmetric

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
LOADCYN       14     0.1                     1.0      51
.
.

Example

In this example, cyclic symmetric analysis will be demonstrated for the following model which consists of shell elements and RBE2s. The basic segment is chosen to be the smallest repeating unit in the model and is one-eighth of the full model (shown in blue).

The base of the geometry is subjected to fixed supports and a pressure loading is applied on the top face.

The CYJOIN definitions are such that the grid points on each side match with the grid points on the other side. This homologous relationship between the grid points is important and incorrect results could be obtained if this condition is violated.

The results from the cyclic symmetry analysis have been compared with a full model as a reference. The results show good agreement.

Comments

HyperMesh support for cyclic symmetry analysis would be available in future releases.