3.4.4.      Iterative Guyan Reduction

Selecting TAM DOF using the IGR method is performed using the iter_guyan alter in Nastran. There are two ways to run this procedure. For fairly small models (<10,000 DOF) it may be possible to start the procedure with no preliminary reduc­tion and allow the algorithm to pick from all DOF in the model. However, for most practical problems it is not possible to do this since the preliminary reductions will require too much memory and disk space[18]. Instead, a candidate DOF set should be provided. This candidate set can be identified using either engineering judgment or a combination of the GPKE and RKE methods outlined in the previous two sections.

The iter_guyan alter will optionally calculate the pseudo-orthogonality matrix listed in Equation 2.4-1 and report two norms on the difference between this matrix and the identity matrix. The first norm (NORT) is the Frobenius norm, which is defined as the square root of the sum of the squares (RSS) of the elements of a matrix. The second norm (MORT) is the maximum absolute value of any element in the matrix. These error norms can be plotted against number of DOF to develop an understanding of TAM accuracy as a function of number of retained DOF.

The IGR method is applied in Nastran by using the iter_guyan alter. Requirements for the use of this alter are as follows:

·         File Assignment Deck. The alter will optionally calculate pseudo-orthogonality of the FEM modes for the candidate set and both pseudo- and cross-orthogonality for the final set.  If this is desired, the FEM mode shapes must be made available using the write_modes alter. If the OUTPUT2 or OUTPUT4 options are chosen, the corresponding file must be assigned using an INPUTT2 or INPUTT4 assignment in the file assignment deck.

·         Executive Control Deck. The iter_guyan alter must be included before the CEND card.

·         Case Control Deck. A METHOD card should be included if the pseudo- and cross-orthogonality are desired.

·         Bulk Data Deck. The bulk data deck must include the standard information for performing a normal modes analysis including ASET cards to define the candi­date DOF if desired. If pseudo- and cross-orthogonalities are desired and the DMIG option was chosen for the FEM mode shapes, these must be included in the bulk data. Otherwise the OMODES parameter must be set as follows:

OMODES < 0            Read the modal frequencies and shapes in OUTPUT2 for­mat from FORTRAN UNIT |OMODES|

OMODES > 0            Read the modal frequencies and shapes in OUTPUT4 for­mat from FORTRAN UNIT OMODES

The IGR method is controlled by the following parameters:

NGUYAN                 Number of DOF to reduce matrices to (no default)

GUFILT                    Filter to eliminate DOF at each iteration (default = 1.0)

ENORM = ‘YES’      Calculate error norms at each step

ENORM = ‘NO’        Do not calculate error norms at each step (default)

The number of DOF eliminated at each step in the IGR process is controlled by the GUFILT parameter. By default this is set to 1.0, and only one DOF will be elimi­nated at each step. If it is set to a value less than 1.0, all DOF with stiffness/mass ratios greater than GUFILT times the maximum stiffness/mass ratio will be elimi­nated at each step[19]. Choosing a value for GUFILT of 0.90 to 0.95 can accelerate the process significantly without greatly changing the final answer.

At the end of the IGR procedure, the retained DOF are written to .f06 file in MATGPR format and the .pch file in DMIG format. An eigensolution is performed using the reduced matrices and, if requested, the pseudo- and cross-orthogonalities are calculated. The solution then stops. Further processing of the reduced set can be performed by converting them to ASET cards and starting a new Nastran run.

No intermediate results are stored during the IGR procedure. If intermediate results are desired[20], the IGR procedure can be run in two steps. For example, if the user is starting with a candidate set of 3,000 DOF and would like to save the DOF associ­ated with the best 1,000 DOF as well as the best 200 he can first run the IGR proce­dure with NGUYAN set to 1,000, convert those 1,000 DOF to ASET cards, and then restart the procedure to reduce it to 200.  By using the ENORM parameter the error as function of number of DOF can be plotted and a suitable stopping point can be chosen.

A sample IGR run is illustrated in Figure 3-18. In this run the initial set is selected to be all translational DOF for every grid in the model that is not restrained. The goal is to choose the best 20 DOF, using a filter of 1.0 so that only one DOF is removed per iteration.

gpsc_igr.dat

 

ASSIGN INPUTT4='gpsc_fem.op4', UNIT=13

ASSIGN MASTER ='gpsc_igr.MASTER', DELETE

ASSIGN DBALL  ='gpsc_igr.DBALL',  DELETE

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SOL 103 $

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INCLUDE 'iter_guyan.v2001'

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CEND

TITLE    =GENERAL PURPOSE SPACECRAFT (GPSC)

SUBTITLE =ITERATIVE GUYAN REDUCTION

LABEL    =FIND BEST 20 DOF

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SPC    = 10             $ Constrain booster interface points

METHOD = 70             $ Modes to 70 Hz

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DISP(PLOT) = ALL        $ Recover displacements

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BEGIN BULK

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PARAM   WTMASS  .00259

PARAM   OMODES  13   

PARAM   ENORM   YES  

PARAM   NGUYAN  10   

PARAM   GUFILT  1.0

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EIGRL   70      1.0     70.0

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$  Spacecraft bulk data

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INCLUDE 'gpsc.blk'

INCLUDE 'gpsc.prp'

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$  Large candidate set (138 DOF) to start process

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INCLUDE 'gpsc_cdof.aset'

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ENDDATA

Figure 3-18. Sample input file for Iterative Guyan Reduction (IGR).

Since PARAM, ENORM has been selected, the error norms are calculated at each step in the reduction process, and are plotted in Figure 3-19.  In this case the error norms are low until 25 DOF, at which point they increase to 5% and 4% respectively.

Figure 3-19. Error norms vs. number of DOF for IGR algorithm