3.4.3.      Iterative Residual Kinetic Energy

Two variations of the iterative residual kinetic energy (IRKE) algorithm are provided with TAMKIT.  This first variation iter_rke performs a standard Guyan reduction at each step.  This is applicable to all structures as long as the initial DOF are sufficient to restrain rigid body motion, but can be very computationally intensive for large models.  The second variation fast_iter_rke implements the alternate method described in Section 2.5.3.  This method is much faster because the large stiffness matrix  is only decomposed once, and the matrix solution involved in each step only requires a forward backward substitution (FBS) to calculate additional columns of, and decomposition of an A-set size matrix .  Because this method requires the initial decomposition of the F-set stiffness matrix it does require a grounded structure.  If the structure is free-free, soft springs can be added to eliminate the zero frequency rigid body modes.  The fast IRKE algorithm should be selected for most models.

The number of DOF added at each step is controlled by the parameter RKFILT. The residual kinetic energy vector is normalized to a maximum value of 1.0, and all DOF with values equal to or above RKFILT are added.  The default value of RKFILT adds one DOF at each step. A lower value can be used to accelerate the algorithm by adding more than one DOF at each step. For example, a value of 0.95 will add all DOF will RKE values above 95% of the maximum at each step. This can vary from a single DOF to ten or more.

Both versions of the 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 IRKE algorithm is implemented in Nastran by using either the iter_rke or the fast_iter_rke’ alters. The differences between the two alters is in the internal algorithm. Because these alters are functionally similar but iter_rke is only practical for very small problems, support for iter_rke will be discontinued after NX Nastran 11.

They have similar inputs and should give the similar results, although the fast_iter_rke alter has a few additional options. Requirements for the use of fast_iter_rke is as follows (requirements for iter_rke can be found in the header of the alter itself):

·         File Assignment Deck. If the FEM modes were stored in either OUTPUT2 or OUTPUT4 format, the corresponding file must be assigned to a FORTRAN Unit Number.

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

·         Case Control Deck. The case control deck must include standard commands for a normal modes analysis (MPC, SPC, DYNRED, and METHOD selections as needed). For the fast_iter_rke alter standard Nastran OMODES output request can be used to limit the FEM modes used in the DOF selection.

·         Bulk Data Deck. The bulk data deck must include the standard information for performing a normal modes analysis including ASET cards to define the initially selected DOF. If the user desires that the retained DOF be chosen from a “candidate” set, these should be identified on USET,U1 cards.  Alternatively, the user can exclude DOF from consideration using USET,U2 cards. If the DMIG option was chosen for the FEM mode shapes, these must be included in the bulk data. Otherwise the MODUNIT parameter must be set as follows:

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

MODUNIT =0            Read the modal frequencies from DMI with name FLAMA and and shapes from DMIG with name UGFEM

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

The following parameters control the execution of the IRKE alters:

WRKE = 'YES’         Write RKE vectors to UNIT 12

WRKE = 'NO’          Do not write RKE vectors to UNIT 12 (default)

WORTHO= 'YES’   Write PO error to OUTPUT4 UNIT 11 at every step

WORTHO= 'NO’    Do not write PO error to OUTPUT4 (default)

ENORM = ‘YES’      Calculate error norms at each step (default)

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

RKFILT                    Select DOF with RKE values > RKFILT (default = 1.0)

NDOF                       Number of retained DOF to stop algorithm (default=1000)

OGOALU1 = 'YES' Use U1 set as orthogonality goal

OGOALU1 = 'NO'   Use identity as goal (default)

 

An example of an IRKE run is illustrated in Figure 3-18. This runs starts with just two triaxial accelerometers at grids 18 and 40 as illustrated in Figure 3-10 and adds DOF up to a total of 50 based on the RKE of FEM modes 1-10.  Error norms are calculated at each step, and the user has supplied a set of “candidate” DOF identified on USET,U1 cards (gpsc.uset).  The final set is limited to a subset of the “candidate” set.

 

gpsc_irke.dat

 

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

ASSIGN MASTER ='gpsc_aset.MASTER', DELETE

ASSIGN DBALL  ='gpsc_aset.DBALL',  DELETE

$

SOL 103 $

$

DIAG 8 $

$

INCLUDE 'fast_iter_rke.v2001'

$

CEND

TITLE    =GENERAL PURPOSE SPACECRAFT (GPSC)

SUBTITLE =ITERATIVE RESIDUAL KINETIC ENERGY ALGORITHM

$

SPC    = 10             $ Constrain booster interface points

METHOD = 70             $ Modes to 70 Hz

$

DISP(PLOT) = ALL        $ Recover displacements

$

$ Only use FEM modes 1-10 for DOF selection

SET 1 = 1 THRU 10

OMODES = 1

$

BEGIN BULK

$

PARAM   WTMASS  .00259

PARAM   MODUNIT  13

PARAM   ENORM   YES

PARAM   NDOF    50

PARAM   RKFILT  0.95

$

EIGRL   70      1.0     70.0

$

$  Spacecraft bulk data

$

INCLUDE 'gpsc.blk'

INCLUDE 'gpsc.prp'

$

$  Initial DOF to start selection process

$

INCLUDE 'gpsc_init.aset'

$

INCLUDE 'gpsc.uset'

$

ENDDATA

Figure 3-15. Sample input file to calculate iterative residual kinetic energy (IRKE).

If PARAM,ENORM,YES, the alter writes the initial pseudo-orthogonality matrix to both the .f06 and .pch files. At each step it also writes the selected DOF to both the .f06 and .pch files and if PARAM,ENORM,YES writes the two error norms (NORT and MORT) to both the .f06 and .pch files. A portion of the .pch file for this run is illustrated in Figure 3-18. ASET cards can be written from the DOF selected by the IRKE algorithm using the pch2aset or evaluate_irke_dof Matlab functions.

 

gpsc_firke.pch

 

DMI     NORT           0       6       1       0               1       1

DMI*    NORT                           1               1  2.27057052E+00

DMI     MORT           0       6       1       0               1       1

DMI*    MORT                           1               1  9.99905050E-01

DMIG    VZG            0       9       2       0                       1

DMIG*   VZG                            1               0

*                     32               3 9.951227863D-01

*                     50               3 1.000000000D+00

DMI     NORT           0       6       1       0               1       1

DMI*    NORT                           1               1  1.77793646E+00

DMI     MORT           0       6       1       0               1       1

DMI*    MORT                           1               1  9.87834811E-01

DMIG    VZG            0       9       2       0                       1

DMIG*   VZG                            1               0

*                     11               3 9.999999974D-01

*                     12               3 9.989132417D-01

*                     13               3 1.000000000D+00

*                     17               3 9.989132451D-01

DMI     NORT           0       6       1       0               1       1

DMI*    NORT                           1               1  6.13082826E-01

DMI     MORT           0       6       1       0               1       1

DMI*    MORT                           1               1  4.59522247E-01

Figure 3-16. Portion of .pch file from IRKE run

The first values of NORT (2.27) and MORT (1.00) are the error norms for the initial set of 6 DOF. At the first step DOF 32-T3 and 50-T3 are added, reducing the error norms to 1.78 and 0.99 respectively. The next step adds DOF 11-T3, 12-T3, 13-T3 and 17-T3, reducing the error norms to 0.61 and 0.46 respectively. The error norms vs. number of DOF are illustrated in Figure 3-17. Both error norms drop rapidly to about 4% error at 16 DOF. The maximum error drops below 2% at 20 DOF and below 1% at 24 DOF.

Figure 3-17. Error norms vs. number of DOF for IRKE algorithm