Elastic representation as superelements

SDT-nlsim Contents Functions

2.4 Elastic representation as superelements

2.4.1 Generic representations in SDT

For interfacing with external finite element software the well documented superelement formalism is used. This formalism is largely used by Multibody Dynamic Software (Simpack, Adams, Excite, ...) and thus widely documented.

A superelement representation of the model is of the form

(2.5)

In general, the reduction is performed so that the DOFs retained {q_R} are related to the original DOFs of a larger model by a Rayleigh Ritz reduction basis T using

(2.6)

This representation is fairly standard. The data structure representation within SDT is described in section ??. SDT/FEMLink supports import from various FEM codes and more details are given in section ?? for NASTRAN and section ?? for Abaqus.

For Craig-Bampton type reduction (enforced displacement on a set of interface DOF I and fixed interface modes on other DOF C), the reduction basis has the form

(2.7)

which verifies the constraints on basis columns that T_II=I and T_IC=0.

For the free mode variant (McNeal) (see sdtweb('fe_reduc','free')), the form is

(2.8)

with no T_I columns.

The observation formalism of SDT which is applicable to both test/analysis sensors (see sdtweb('sensor')) and strain observations used for non-linearities .

(2.9)

Standard notions that have an equivalent in other code are

q_I interface DOF are directly comparable in SDT and other software when using a DofSet entry that contains an identity enforced motion matrix on a set of DOF idof, typically known as MASTER degree of freedom. That can be initialized with a call of the form model=fe_case(model, 'DofSet', 'In',struct('DOF',idof, 'def', speye(length(idof)))).
T_C columns may correspond to generalized or internal DOFs. External software will often require that a DOF number be associated to these interfaces. xxx
b_res the independent vectors used to generate residual loads are found as columns of a DofLoad entry. For point loads on a set of DOF adof the entry would be struct('DOF',adof, 'def',speye(length(adof))).
[c] observation matrices do not exist in other environments. The strategy usually retained for export is to add additional nodes of a set ObsNode to correspond to the observations of interest and define the observation equation using a multiple point constraint. This is achieved by modifying the model using fe_sens('MeshSensAsMPC'.
rigid assuming that the 3D motion of all nodes in the set depend rigidly on the center node motion. Can be used to define motion of sensor nodes. This tends to over-stiffen the area of connected nodes.
rbe3 assuming that the center node moves as the mean motion of the set of nodes is often considered to observe motion of nodes. The case dependent observations of SDT are more general, but may correspond to rbe3.

Implementations of these are discussed for Abaqus ??, NASTRAN ??, ANSYS ??.

2.4.2 NASTRAN cards used for sensors/non-linearities

The NASTRAN equivalent of superelement notions discussed in section ?? are

q_I interface DOF of are defined in NASTRAN using Bset cards. These are stored in SDT as a DofSet entry to the model.
b_res the independent vectors used to generate residual loads and lead to additional shapes using the residual vector procedures of NASTRAN
- point loads simply declared using the USET,U6 card
- relative loads simply can obtained by declaring a CDAMP element that generates a relative viscous load between its two nodes.
- [b] is defined by an DAREA real loading and possibly DPHASE definition. It should be noted that in SDT, it is strongly advised to define the phase using the input, since a complex input shape matrix has no sense in the time domain. The input is defined using a RLOAD2 B(f)e^{iφ(f)+θ−2π f τ} or RLOAD1 (C(f)+iD(f))e^{iθ −2π f τ}
T_C columns are associated to scalar DOFs called QSET. These require the definition of a QSET card (to declare existing DOFs), SPOINT grids (to have node numbers to support these QSET DOFs). Note also that the SPOINT numbers should be distinct from other NodeId. The number of modes defined in the EIRGL card should be lower than the the number of SPOINT and the QSET card.
y=[c]{q} observation. Does not exist in NASTRAN documentation, but implemented exporting SPOINT for each observation component and an MPC for each row of the observation matrix. This is achieved by modifying the model using fe_sens('MeshSensAsMPC' prior to export.
rigid is known as RBE2 in NASTRAN.
rbe3 is known as RBE3 in NASTRAN.

Laws without internal states are similar to PGAP and import will be implemented in the future.

2.4.3 ABAQUS cards used for sensors/non-linearities

The Abaqus equivalent of superelement notions discussed in section ?? are

q_I interface DOF xxx
b_res the independent vectors used to generate residual loads are written as independent load cases using xxx SeWriteBres
```
*LOAD CASE, NAME=LC000001
*CLOAD, OP=MOD
1001, 1, 1
*END LOAD CASE
```
xxxGV resvec,
*EQUATION : equivalent the SDT MPC definition with a direct constraint matrix declaration.
*KINEMATIC COUPLING : equivalent of SDT rigid connections where the spring is connected to a master node with 6 DOF which enforce motion of a number of slave DOFs.
*DISTRIBUTING COUPLING equivalent of SDT RBE3 : flexible connection where the spring is is connected to a slave node with 3 or DOF which depend from a set of master nodes.
*COUPLING : specific surface based definition, followed by either a *KINEMATIC card for rigid or *DISTRIBUTING card for RBE3 formulations.
*MPC : node based definition with type BEAM to constraint 6 DOF per node or type PIN to constraint the 3 translations only.
*CONNECTOR : connectors provide advanced structural kinematics, type BEAM without elasticity definition provides a rigid connection (linearized in SDT).

2.4.4 ANSYS cards used for sensors/non-linearities

For spring representations of volumes or surfaces, a first common approach is to use so called rigid elements. ANSYS supports

CE, CERIG, MPC184, RBE 2 : rigid connections where the spring is connected to a master node with 6 DOF which enforce motion of a number of slave DOFs.
TARGE 170+CONTA 173, TARGE 170+CONTA 174

2.4.5 NASTRAN Craig-Bampton example

The superelement generation by NASTRAN is saved to an .op2 file that is automatically transformed to the SDT superelement format by FEMLink. A sample file is given in ubeamse.dat.

ASSIGN OUTPUT2='./ubeam_se.op2',UNIT=30
$
ID DFR
SOL 101
GEOMCHECK NONE
TIME 100
$
CEND
TITLE=Generic computation of mode shapes
METHOD=1
DISP(PLOT) = ALL
SPCFORCES(PLOT)=ALL
MPCFORCES(PLOT)=ALL
$ Now extract stresses on base
SET 101=1 THRU 16
STRESS(SORT)=101
$
MPC=1
SPC=1
$ RESVEC(NOINRL)= YES
EXTSEOUT(ASMBULK,EXTBULK,EXTID=100,DMIGOP2=30)
PARAM,POST,-2
PARAM,BAILOUT,-1
$
BEGIN BULK   
$EIGRL,SID,V1,V2,ND,MSGLV,MAXSET,SHFSCL,NORM
EIGRL,1,,,20
$ DOF and nodes to support modal DOF
QSET1,0,1000001,THRU,1000050 
SPOINT,1000001,THRU,1000050
$ Master DOF 4 base corners
BSET1,123,1,5,8,12
$ 
$ Residual on 3 DOF of input node 104
USET,U6,104,123
$
$ Residual associated with CAMP1 vector
CDAMP1  161     2       114     1       244     1
PDAMP*  2               1.              
$
include 'ubeam_include.bdf'
ENDDATA

The resulting basis has the following form

[T]=[

I	0	0
−K_CC⁻¹K_CI	[φ_C]_1:NM	[K_CC⁻¹ [b_res]]_⊥

] (2.10)

The op2 file contains nodes and superelement definition. It is advised to read the bulk file to obtain a model containing elements and material properties.

The interface DOF are defined in NASTRAN usingBset cards. These are stored in SDT as a DofSet entry to the model.
QSET correspond to modal/generalized DOFs. These require the definition of a QSET card (to declare existing DOFs), SPOINT grids (to have node numbers to support these QSET DOFs). Note also that the SPOINT numbers should be distinct from other NodeId. The number of modes defined in the EIRGL card should be lower than the the number of SPOINT and the QSET card.
Fixed interface modes φ_c are computed by specifying the EIRGL card.
Residual loads b_res are defined as follows and lead to additional shapes using the residual vector procedures of NASTRAN
- point loads simply declared using the USET,U6 card
- relative loads simply obtained by declaring a CDAMP element that generates a relative viscous load between its two nodes.

2.4.6 Free mode using NASTRAN

When using a free mode computation, NASTRAN provides mechanisms to compute residual vectors, you should just insert the RESVEC=YES card. An example is given in the ubeamfr.dat file. The resulting basis has the following form

[T]=[

[φ]_1:NM

[K_Flex⁻¹ [b_res]]_⊥

] (2.11)

The main mechanisms to generate residual vectors are

Free interface modes φ_c are computed by specifying the EIRGL card.
Residual loads b_res are defined as follows and lead to additional shapes using the residual vector procedures of NASTRAN
- point loads simply declared using the USET,U6,NodeId,DofList card. Alternatively RVDOF (MSC but possibly not NX-NASTRAN) can given a list of up to four NodeId,Dof per card.
- relative loads simply obtained by declaring a CDAMP element that generates a relative viscous load between its two nodes.

2.4.7 Storing advanced SDT options in bulk format

For upcom parameters, export is done using design variables.

SDT-NLSIM provides an harmonic definition mechanism (see hdof). Storage in NASTRAN bulk format is as follows

$   1  $$   2  $$   3  $$   4  $$   5  $$   6  $$   7  $$   8  $$   9  $
$ All DOFs with sin(omega t) and cos(omega t) 
DTI     HDOF    1       ALL     123456  CS1     ENDREC
$ Gradual building of full list of DOFs 
DTI     HDOF    1       N1      THRU    N2      123     S1      N3
        THRU    N4      1       C1      N5      123456  S1      ENDREC

Node numbers are first specified using ALL all (independent) nodes, N1 THRU N2 a list of consecutive node numbers, N5 a single node number. Associated DOFs are then written using the CM field of RBE2 (Component numbers of the dependent degrees-of-freedom integers 1 through 6 with no embedded blanks). A third field then specifies the harmonics. cs1 is a short cut for both cos(1 ω t) and sin(1 ω t).

The specification of target frequencies follows the normal NASTRAN format using FREQ or FREQ1 cards. Provision for a single call generating responses at multiple amplitudes (hbm_solve AFMap .Freq and .Amp fields) is specified as a DTI HBMAmp entry with all target amplitudes given.

$   1  $$   2  $$   3  $$   4  $$   5  $$   6  $$   7  $$   8  $$   9  $
EIGRL,10,,,1
$FREQ,SID,F2,F2,F3
$FREQ1,SID,F1,DF,NDF
FREQ          10   0.318      1.     3.0     4.0
RLOAD1        10       1                       1

$   1  $$   2  $$   3  $$   4  $$   5  $$   6  $$   7  $$   8  $$   9  $
DTI     HBMAmp  1       1.0     2.0     3.0     ENDREC

To specify loads, a number of formats are defined.

$   1  $$   2  $$   3  $$   4  $$   5  $$   6  $$   7  $$   8  $$   9  $
DTI     Name    1       SID     101     FORM    Amp     UN1     UN2    
        Harmi   ACi     ASi     ENDREC

Name is an arbitrary string (at most 8 characters) but should be unique and differ from internal NASTRAN tables. By default it is proposed to use strings of the form P101 where 101 is the property number.
IREC (field 3 of the DTI) is only used when considering multiple entries with the same name and should be set to 1.
SID : first the string SID in field 4 then, in field 5, the property identifier (integer) which should correspond to the set identification number SID for which this amplitude dependence is defined.
Form (selected with the string on field 7) is the form name with the following formats defined
Form

Amp amplitudes {u(t)} = C₀+∑_{k∈ H} S_k sin(kω t) + C_k cos(kω t)

AmpT Amp table {u(t)} = C₀(ω)+∑_{k∈ H}S_k(ω) sin(kω t) + C_k(ω) cos(kω t)
Harmi number of retained harmonic. 1 for cos(1 ω t) and sin(1 ω t).
ACi, ASi amplitudes associated with the cosine and sine harmonic contributions. In the AmpT form integer numbers referring to table entries in the bulk.

2.4.8 Abaqus Craig-Bampton example

Superelement generation in Abaqus is divided in three steps.

*STEP, PERTURBATION//*STATIC used to define residual vectors. Note that export of residual loads associated with non-linearities is not yet implemented in SDT.
*FREQUENCY, EIGEN=LANC to compute internal modes with possibly a Craig-Bampton interface declared by a *BOUNDARY card.
*SUBSTRUCTURE GENERATE to generate and export the superelement, use *RETAINED NODAL DOF associated to fixed DOF in the frequency step for a Craig Bampton reduction.

 See SeGenResidual.inp

2.4.9 Abaqus McNeal example

xxx

2.4.10 ANSYS Craig-Bampton example

The cards typically used for superelement generation in ANSYS are

antype,substr specify FE substructure generation in after /SOLU
cmsopt,fix,Nshapes,,,,,tcms card generates .sub. Use ans2sdt('subSE','file.sub') to import matrices from .sub, restitution from .tcms and mesh from .cdb files using the same file root.
resvec,on to use residual vectors in the basis
seopt,name,MatType,1 with MatType=2 for mass and stiffness, and 3 for stiffness, mass, viscous damping, name must be defined with card /FILENAME before the analysis.
m,NodeId,all master DOF definition repeat card for the various interface nodes. You will have to replace all with UX,,UY,UZ,ROTX,ROTY,ROTZ if the DOF is used by an element that supports multi-physics.

/FILENAME,ubeam_se  ! name must be the one used in SeOpt command
/PREP7
!...
! Use command F to apply loads that will define the residual vector
/SOLU

antype,substr  ! substructure analysis

CmsOpt,Fix,20,,,,,TCMS
RESVEC, ON
SeOpt,ubeamse_ans,3,1,0,,  ! 3(all matrices,2 for m and k), 1 to print

! Define list of master DOF, you cannot use ALL if the elements support multi-physics
M,1,UX,,,UY,UZ,ROTX,ROTY,ROTZ
M,5,UX,,,UY,UZ,ROTX,ROTY,ROTZ
M,8,UX,,,UY,UZ,ROTX,ROTY,ROTZ
M,12,UX,,,UY,UZ,ROTX,ROTY,ROTZ

SAVE  ! save .db file
SOLVE ! generate the matrices
FINISH

Form
Amp	amplitudes	{u(t)} = C₀+∑_{k∈ H} S_k sin(kω t) + C_k cos(kω t)
AmpT	Amp table	{u(t)} = C₀(ω)+∑_{k∈ H}S_k(ω) sin(kω t) + C_k(ω) cos(kω t)