Craig-Bampton Reduction

5. Craig-Bampton Reduction#

The CBR solution method makes a superelement as specified in the CBModel section of the text input file. The requirements for Sierra/SD to use this superelement are in the next section. This reduction is often called a Component Mode Synthesis (or CMS). For limitations and use cases see the Craig-Bampton reduction Solution Case section of the Users manual.

5.1. Input Required#

The following input is required to run the CBR solution.

5.1.1. Solution#

The solution section must contain input for the number of modes. This is the number of fixed interface modes to compute. It must be entered, and will be different than the number of system modes desired. It must also contain shift to ensure that the matrices are not indefinite. See the Craig-Bampton reduction Solution Case section of the Users manual for more details on full list of solution parameters.

5.1.2. CBModel#

The CBModel defines most of the parameters for the solution. It defines the interface boundary nodes. Note that all degrees of freedom of each node is a part of the model. Either define all six degrees of freedom as interface dofs, or permit them to be reduced in this step. Interface nodes may be connected to any structural element (solids, shells or beams), but not to a constraint relation.

For example,

CBMODEL
  nodeset=1
  format=mfile
  file=cbr.m
  GlobalSolution = yes
  inertia_matrix = yes
END

See the example below for helpful insights on how to verify the model is a good approximation of the original system.

5.1.3. History#

For the CBR solution case, the history file contains the Output Transfer Matrix (OTM). The history section is only necessary if the OTM output is desired. Otherwise, it is optional. Only the following will be honored (others will be ignored).

displacement
strain
stress

Note that the transfer matrices for acceleration and velocity are obtained by differentiating the displacement equation.

5.2. Mesh#

The geometry as shown in Figure 5.1 consists of a cone with a nodeset on the top and bottom edge. The model reduction consists in reducing the stiffness matrix from the 80 nodes in this model to the interface nodes (3 nodes on the base in nodeset 3). Thus, there are 18 constraint modes. We choose to retain 4 fixed interface modes for this example.

5.3. Example#

Running the model and examining the output, you will notice the following.

For this example there are two sets of eigenvalues (Ritz values) output to the screen. The first, a set of 10 modes, corresponds to the eigen problem of the unreduced model which includes 6 zero energy modes. The second set of modes is the fixed interface modes of the analysis. The first \(4\) modes in CBR-CBR.exo correspond to these fixed interface modes.
The result file, CBR.rslt, contains three sets of eigenvalues; the two mentioned above and the eigenvalues of the reduced system. No eigenvectors from the reduced system can be output since there is no geometry database associated with it. The last set of eigenvalues includes every eigenvalue of the reduced system.

Notice also that the eigenvalues of the reduced system are not identical to the unreduced system. However, even with only four fixed interface modes, the first elastic mode agrees up to the 4th digit. General practice would ensure that the maximum frequency of the fixed interface modes is at least twice the frequency of interest.
The cbmap is found in both the result file and the reduced model output file. This map relates rows and columns of the reduced system with physical quantities. The first of the \(3\) nodes in the nodeset has global id \(1\) as shown in the figure. All \(6\) degrees of freedom are active at each node. And the cbmap has \(18\) rows.
The reduced system is \(22\) degrees of freedom, which consists of \(4\) fixed interface modes and \(18\) constraint modes (\(6\) degrees of freedom associated with \(3\) nodes). The mass and stiffness matrices are almost full. Generally, the constraint modes contribute full matrix terms to both mass and stiffness.
Rerunning with mfile added to the output section creates many files that will not be described here including the \(\Phi\) and \(\Psi\) matrices.

The output is written to the file CBR.m. Extracts from this file are included below from which you note the following.

All the data required for the model reduction is found in a single file.
The map of the reduced model is defined in cbmap. A map of the output transfer matrix rows is OutMap.
There are always 6 degrees of freedom per node in the OutMap. This example does not show this, but there may be fewer in the cbmap. Note that while Kr and Mr are reduced system matrices which must be nonsingular, OTM is a transfer matrix and can include inactive degrees of freedom.

NumC=18;
NumEig=4;
Kr=zeros(22,22);
Kr(1,1)=7.703363317234302e+04;
Kr(2,2)=9.043236930586677e+04;
...

Mr=zeros(22,22);
Mr(1,1)=1.000000000000000e+00;
Mr(1,5)=-9.545115933105166e-03;
...

% map of nodes in the output transfer matrix
% OutMap is the global node number
% There are exactly 6 outputs per node.
OutMap=zeros(1,32);
OutMap=[1 5 6 10 11 15 16 20 21 25 26 30 31 35 36 40 41 45 ...
OTM=zeros(192,22);
OTM(1,5)=1.000000000000000e+00;
OTM(2,6)=1.000000000000000e+00;
...

%cbmap(:,1) is global node id (1:n)
%cbmap(:,2) is coordinate (x=1, y=2, etc.)
%the first 4 dofs in the matrices are modes,
% while the last 18 dofs are interface dofs.
cbmap=[1 1
1 2
1 3
1 4
1 5
1 6
...

5.4. Verification of the Model#

The following are some things that can be done to ensure that the model has been properly developed.

5.4.1. Comparison of Reduced and Full Eigenvalues#

It is a very good idea to compare the eigenvalues of the full and reduced system. It will approximately double the computational effort of the model reduction, but there is very little set up time. The example does this. All that is required is to compute the results in a multi-case approach. Begin by computing the eigenvalues of a full system. Then, in the next case compute the reduced order model. By including GlobalSolution in the CBModel section, the eigenvalues of the reduced system are also computed. These eigenvalues and frequencies appear in the text result file, under the heading Eigenvalues of Reduced System.

5.4.2. Comparison of Reduced and Full Displacements#

It is significantly more complicated to compare the displacements of the two models because there is no automatic upstream data recovery. Manual data recovery will have to be done in MATLAB. We illustrate the method with a small transient run, but it could also be done for a eigen analysis (or statics if the model is statically determinant).

Consider a calculation of 2000 time steps each of \(10^{-5}\) seconds. We impulsively load the structure on the interface (nodeset 3) with a force in the \(y\) direction only. The load begins at zero, ramps to 10\(^6\) at \(10\,\mu s\), and then ramps back to zero at \(20\,\mu s\). Output will be examined on nodesets 1 and 2. This example is found in CBR_trans.inp.

Following the calculation, data from any of the output nodes can be evaluated using the history file. The following commands evaluate the \(x\) displacement on node 70.

unix% exo2mat CBR-transient.h
unix% matlab
load CBR-transient
k=find(node_num_map==70);
plot(time,nvar01(k,:));

The reduced model can be used to perform the same calculation. The MATLAB commands to do this work once CBR.m has been read into MATLAB are included here.

nsteps=2000;
ff=zeros(1,nsteps);
ff(2)=1;
neq=max(size(Kr));
force=zeros(neq,1);
rows=NumEig + find(cbmap(:,2)==2);
force(rows)=1e6;
dt=1e-5;
u=CBRint(Kr,Mr,force,ff,dt);
time=(1:nsteps)*dt;
k=find(OutMap==70);
orow=(k-1)*6+1;  % x component of node 70
U70x=OTM(orow,:)*u;

The time integration is a standard Newmark integration performed using CBRint.m, which is available in the test directory.

Finally, we can compare the results, which are shown in Figure 5.2. The data in the figure is obtained by running the CBR reduction with a varying number of fixed interface modes. Note that 4 modes, and even 10 modes are not sufficient to capture the gross response of the structure at node 70. Even at 50 modes there is high frequency data that has been lost. This is as expected since the reduced model is designed to capture only the low frequency response of the structure. The first elastic mode at 21 Hz has a period of 48 ms.

Figure 5.2 Example CBR transient computations.#

5.5. What to do with the Results#

5.5.1. solving the system#

The reduced mass and stiffness matrices contain the dynamics of the system. These could be solved in an eigen analysis for example in MATLAB.

[][e_value,e_vector]=eig(Kr,Mr);

The eigenvalues, e_value, represent the system natural frequencies. The eigenvectors are a mix of generalized and physical degrees of freedom. The OTM is used to compute the response on the physical degrees of freedom on the nodesets in the history file.

Out=OTM*e_vector;

To find the response on a specific degree of freedom use the OutMap. For example, to find the \(Z\) degree of freedom on node 25 of the model.

index = find(OutMap==25);
k = (index-1)*6 + 3;
for i=1:size(Out,2)
  fprintf('Mode %d, Z value on node 25 = %g\n',i,Out(k,i))
end

When this document was written no process was available to take these results back into an Exodus database so the resulting displacement mode shapes can be plotted on the original model.

5.5.2. Incorporate the reduced model into another system model#

This is one of the more important reasons for doing a model reduction. The approach depends on the format of the new model. The following are options.

Sierra/SD.

Sierra/SD can input a CBR model in netcdf format as a superelement. See Section 6.
MATLAB.

The model can be combined with other models in MATLAB. The trick is to use the cbmap to tie together different degrees of freedom.
NASTRAN.

NASTRAN can do this.