# 6.3 Superelements and CMS#

## 6.3.1 Superelements in a model#

A superelement is a model that is included in another global model as an element. In general superelements are reduced: the response at all DOFs is described by a linear combination of shapes characterized by generalized DOFs. The use of multiple superelements to generate system predictions is called Component Mode Synthesis (CMS). For a single superelement (SE structure not included in a larger model) simply use fe_reduc calls. This section addresses superelements integrated in a model.

Starting with SDT 6, superelements are stored as SE entries in the model stack (of the form 'SE', SEname, SEmodel) with fields detailed in section 6.3.2. Superelements are then referenced by element rows in a group of SE elements in the global model. A group of superelements in the Elt matrix begins by the header row [Inf abs('SE') 0]. Each superelement is then defined by a row of the form

[NameCode N1 Nend BasId Elt1 EltEnd MatId ProId EltId].

- NameCode is an identifier encoding the superelement name using fesuper('s_name'). It is then assumed that the model stack contains an 'SE',name entry containing the model constituting the superelement. The encoding uses base2dec and is limited to 8 alphabetic lower case characters and numbers, you can use NameCode = feval(fesuper('@cleanSEname'),NameCode); to test the name compatibility.
- [N1 Nend] and [Elt1 EltEnd] are ranges of implicit NodeId and EltId of the superelement nodes and elements in the global model. That is to say that each node or element of the superelement is identified in the global model by an Id that can be different from the original Id of the superelement model stored in the stack. For more details see Node.
- BasId is the basis identifier in the bas field of the global model. It allows re-positioning of the superelement in the global model.
- Elt1,EltEnd give the range of EltId used to identify elements constituting the superelement. These numbers are distinct from the superelement identifier itself.
- MatId,ProId,EltId are used to associate properties to a given superelement. Superelements support p_super property entries. Material information can be used for selection purposes.

The d_cms demo illustrates the Component Mode Synthesis based on a superelement element strategy. The model of this example (shown below) is composed by two stiffened plates. CMS here consists in splitting the model into two superelement plates that will be reduced, before computation of the global model modes.

- Run step 1 builds the simple model shown above
- Run in step 2 the two parts are separated and defined as super-elements
- Run now display

Other examples of superelement use are given in section 6.3.3.

## 6.3.2 SE data structure reference#

The superelement data is stored as a 'SE',Name,Data entry of the global model stack. The following entries describe standard fields of the superelement Data structure (which is a standard SDT model data structure with possible additional fields).

## Opt#

Options characterizing the type of superelement as follows:

Opt(1,1) | 1 classical superelements, 3 FE update unique superelements (see upcom). |

Opt(1,4) | 1 for FE update superelement uses non symmetric matrices. |

Opt(2,:) | matrix types for the superelement matrices. Each non zero value on the second row of Opt specifies a matrix stored in the field K{i} (where i is the column number). The value of Opt(2,i) indicates the matrix type of K{i}. For standard types see MatType. |

Opt(3,:) | is used to define the coefficient associated with each of the matrices declared in row 2. An alternative mechanism is to define an element property in the il matrix. If these coefficients are not defined they are assumed to be equal to 1. See p_super for high level handling. |

## Node#

Nominal node matrix. Contains the nodes used by the unique superelement or the nominal generic superelement (see section 7.1). The only restriction in comparison to a standard model Node matrix is that it must be sorted by NodeId so that the last node has the largest NodeId.

In the element row declaring the superelement (see above) one defines a node range N1 NEND. The constraint on node numbers is that the defined range corresponds to the largest node number in the superelement (NEND-N1+1=max(SE.Node(:,1))). Not all nodes need to be defined however.

Nodes numbers in the full model are given by

NodeId=SE.Node(:,1)-max(SE.Node(:,1))+NEND

N1 is really only used for coherence checking).

## K{i},Klab{i},DOF#

Superelement matrices. The presence and type of these matrices is declared in the Opt field (see above) and should be associated with a label giving the meaning of each matrix.

All matrices must be consistent with the .DOF field which is given in internal node numbering. When multiple instances of a superelement are used, node identifiers are shifted.

## Elt, Node, il, pl#

Initial model retrieval for unique superelements. Elt field contains the initial model description matrix which allows the construction of a detailed visualization as well as post-processing operations. .Node contains the nodes used by this model. The .pl and .il fields store material and element properties for the initial model.

Once the matrices built, SE.Elt may be replaced by a display mesh if appropriate.

## TR#

TR field contains the definition of a possible projection on a reduction basis. This information is stored in a structure array with fields

- .DOF is the model active DOF vector.
- .def is the projection matrix. There is as many columns as DOFs in the reduced basis (stored in the DOF field of the superelement structure array), and as many row as active DOFs (stored in TR.DOF).
- .adof, when appropriate, gives a list of DOF labels associated with columns of TR.def
- .data, when appropriate, gives a list frequencies associated with columns of TR.def
- .LargeDOF can be used to specify DOFs used to track the large rotation of frame where the superelement is defined in multi-body systems.
- .KeptDOF can be used to specify master DOFs not included TR.def but that should still be used for display of the superelement.

## 6.3.3 An example of SE use for CMS#

Following example splits the 2 stiffened plane models into 2 sub models, and defines a new model with those 2 sub models taken as superelements.

First the 2 sub models are built

model=demosdt('Tuto CMSSE -s1 model'); SE1.Node=model.Node; SE2.Node=model.Node; [ind,SE1.Elt]=feutil('FindElt WithNode{x>0|z>0}',model); % sel 1st plate SE1.Node=feutil('OptimModel',SE1); SE1=feutil('renumber',SE1); [ind,SE2.Elt]=feutil('FindElt WithNode{x<0|z<0}',model); % sel 2nd plate SE2.Node=feutil('OptimModel',SE2); SE2=feutil('renumber',SE2);

Then mSE model is built including those 2 models as superelements

mSE.Node=[]; mSE.Elt=[Inf abs('SE') 0 0 0 0 0 0; % header row for superelements fesuper('s_se1') 1 16 0 1 1 100 100 1 ; % SE1 fesuper('s_se2') 101 116 0 2 2 101 101 2]; % SE2 mSE=stack_set(mSE,'SE','se1',SE1); mSE=stack_set(mSE,'SE','se2',SE2); feplot(mSE); fecom('promodelinit')

This is a low level strategy. fesuper provides a set of commands to easily manipulate superelements. In particular the whole example above can be performed by a single call to fesuper('SelAsSE') command as shown in the CMS example in section 6.3.3.

In this example one takes a full model split it into two superelements through element selections

model=demosdt('Tuto CMSSE -s1 model'); % get the full model feutil('infoelt',model) mSE=fesuper('SESelAsSE-dispatch',model, ... {'WithNode{x>0|z>0}';'WithNode{x<0|z<0}'}); feutil('infoelt',mSE) [eltid,mSE.Elt]=feutil('eltidfix;',mSE);

Then the two superelements are stored in the stack of mSE. Both of them are reduced using fe_reduc (with command option -SE for superelement, and -UseDof in order to obtain physical DOFs) Craig-Bampton reduction. This operation creates the .DOF (reduced DOFs), .K (superelement reduced matrices) and .TR (reduction basis) fields in the superelement models.

Those operations can be performed with following commands (see fesuper)

mSE=fesuper(mSE,'setStack','se1','info','EigOpt',[5 20 1e3]); mSE=fesuper(mSE,'settr','se1','CraigBampton -UseDof'); mSE=fesuper(mSE,'setStack','se2','info','EigOpt',[5 20 1e3]); mSE=fesuper(mSE,'settr','se2','CraigBampton -UseDof');

This is the same as following lower level commands

SE1=stack_get(mSE,'SE','se1','getdata'); SE1=stack_set(SE1,'info','EigOpt',[5 50.1 1e3]); SE1=fe_reduc('CraigBampton -SE -UseDof',SE1); mSE=stack_set(mSE,'SE','se1',SE1);SE2=stack_get(mSE,'SE','se2','getdata'); SE2=stack_set(SE2,'info','EigOpt',[5 50.1 1e3]); SE2=fe_reduc('CraigBampton -SE -UseDof',SE2); mSE=stack_set(mSE,'SE','se2',SE2);

Then the modes can be computed, using the reduced superelements

def=fe_eig(mSE,[5 20 1e3]); % reduced model dfull=fe_eig(model,[5 20 1e3]); % full model

The results of full and reduced models are very close. The frequency error for the first 20 modes is lower than 0.02 %.

fesuper provides a set of commands to manipulate superelements. fesuper('SEAdd') lets you add a superelement in a model. One can add a model as a unique superelement or repeat it with translations or rotations.

For CMS for example, one has to split a model into sub structure superelement models. It can be performed by the fesuper SESelAsSE command. This command can split a model into superelements defined by selections, or can build the model from sub models taken as superelements. The fesuper SEDispatch command dispatches the global model constraints (FixDof, mpc, rbe3, DofSet and rigid elements) into the related superelements and defines DofSet (imposed displacements) on the interface DOFs between sub structures.

## 6.3.4 Obsolete superelement information#

The following strategy is now obsolete and should not be used even though it is still tested.

Superelements are stored in global variables whose name is of the form SEName. fe_super ensures that superelements are correctly interpreted as regular elements during model assembly, visualization, etc. The superelement Name must differ from all function names in your MATLAB path. By default these variables are not declared as global in the base workspace. Thus to access them from there you need to use global SEName.

Reference to the superelements is done using element group headers of the form [Inf abs('name')].

The fesuper user interface provides standard access to the different fields (see fe_super for a list of those fields). The following sections describe currently implemented commands and associated arguments (see the commode help for hints on how to build commands and understand the variants discussed in this help).

Warnings. In the commands superelement names must be followed by a space (in most other cases user interface commands are not sensitive to spaces).

- Info Outputs a summary of current properties of the superelement Name.
- Load, Save
Load FileName loads superelements (variables with name of the form SEName) present in the file.

Save FileName Name1 Name2 ... saves superelements Name1, Name2 ... in the file.

Note that these commands are really equivalent to global SEName;save FileName SEName and global SEName;load FileName SEName. - Make elt=fesuper('make Name generic') takes a unique superelement and makes it generic (see fe_super for details on generic superelements). Opt(1,1) is set to 2. SEName.DOF is transformed to a generic DOF form. The output elt is a model description matrix for the nominal superelement (header row and one element property row). This model can by used by femesh to build structures that use the generic superelement several times (see the d_cms2 demo).
make complete adds zero DOFs to nodes which have less than 3 translations (DOFs .01 to .03) or rotations (DOFs .04 to .06). Having complete superelements is important to be able to rotate them (used for generic superelements with a Ref property).

- New New unique superelement declaration using the general format

fesuper ('New Name',FEnode,FEelt). If a superelement called Name exists it is erased. The Node and Elt properties are set to those given as arguments. The Patch property used by feplot for display is initialized.

Set calls of the form fesuper('Set Name FieldOrCommand', 'Value') are obsolete and replaced as follows

- ref field are now replaced by the definition of local bases for each instance of the superelement.
- patch simply replace the superelement .Elt field by another simplified model to be used for viewing once the matrices have been defined.
- ki type fesuper('set Name k i type',Mat) sets the superelement matrix K{i} to Mat and its type to type. The size of Mat must be coherent with the superelement DOF vector. type is a positive integer giving the meaning of the considered matrix (see MatType).

## 6.3.5 Sensors and superelements#

All sensors, excepted resultant sensor, are supported for superelement models. One can therefore add a sensor with the same way as for a standard model with fe_case ('SensDof') commands: fe_case(model, 'SensDof [append, combine] SenType', Name, Sensor). Name contains the entry name in the stack of the set of sensors where Sensor will be added. Sensor is a structure of data, a vector, or a matrix, which describes the sensor (or sensors) to be added to model. Command option append specifies that the SensId of latter added sensors is increased if it is the same as a former sensor SensId. With combine command option, latter sensors take the place of former same SensId sensors. See section 4.7 for more details.

Following example defines some sensors in the last mSE model

% First two steps define model and split as two SE mSE=demosdt('tuto CMSSE -s2 mSE');mSE=fesuper(mSE,'setStack','se1','info','EigOpt',[5 50 1e3]); mSE=fesuper(mSE,'settr','se1','CraigBampton -UseDof'); mSE=fesuper(mSE,'setStack','se2','info','EigOpt',[5 50 1e3]); mSE=fesuper(mSE,'settr','se2','CraigBampton -UseDof');

Sensors={[1,0.0,0.75,0.0,0.0,1.0,0.0]; % Id,x,y,z,nx,ny,nz [2,10,0.0,0.0,1.0]; % Id,NodeId,nx,ny,nz [29.01]}; % DOF for j1=1:length(Sensors); mSE=fe_case(mSE,'SensDof append trans','output',Sensors{j1}); end mSE=fe_case(mSE,'SensDof append stress','output',[111,22,0.0,1.0,0.0]);

fe_case('SensMatch') command is the same as for standard models

mSE=fe_case(mSE,'SensMatch Radius2','output');

Use fe_case('SensSE') to build the observation matrix on the reduced basis

Sens=fe_case(mSE,'SensSE','output');

For resultant sensors, standard procedure does not work at this time. If the resultant sensor only relates to a specific superelement in the global model, it is however possible to define it. The strategy consists in defining the resultant sensor in the superelement model. Then one can build the observation matrix associated to this sensor, come back to the implicit nodes in the global model, and define a general sensor in the global model with the observation matrix. This strategy is described in following example.

One begins by defining resultant sensor in the related superelement

SE=stack_get(mSE,'SE','se2','GetData'); % get superelement Sensor=struct('ID',0, ... 'EltSel','WithNode{x<-0.5}'); % left part of the plate Sensor.SurfSel='x==-0.5'; % middle line of the plate Sensor.dir=[1.0 0.0 0.0]; % x direction Sensor.type='resultant'; % type = resultant SE=fe_case(SE,'SensDof append resultant',... 'output',Sensor); % add resultant sensor to SE

Then one can build the associated observation matrix

SE=fe_case(SE,'SensMatch radius .6','output'); % SensMatch Sens=fe_case(SE,'Sens','output'); % Build observation

Then one can convert the SE observation matrix to a mSE observation matrix, by renumbering DOF (this step is not necessary here since the use of fesuper SESelAsSE command assures that implicit numbering is the same as explicit numbering)

cEGI=feutil('findelt eltname SE:se2',mSE); % implicit nodes of SE in mSE i1=SE.Node(:,1)-max(SE.Node(:,1))+mSE.Elt(cEGI,3); % renumber DOF to fit with the global model node numbers: NNode=sparse(SE.Node(:,1),1,i1); Sens.DOF=full(NNode(fix(Sens.DOF)))+rem(Sens.DOF,1);

Finally, one can add the resultant sensor as a general sensor

mSE=fe_case(mSE,'SensDof append general','output',Sens);

One can define a load from a sensor observation as following, and compute FRFs:

mSE=fe_case(mSE,'DofLoad SensDofSE','in','output:2') % from 2nd output sensordef=fe_eig(mSE,[5 20 1e3]); % reduced model nor2xf(def,mSE,'acc iiplot'); ci=iiplot;