Tech Briefs
Component Mode Synthesis and the Subspace Iteration Method
In this Tech Brief we present some studies of the use of the component mode synthesis^{*} and the subspace iteration method for obtaining eigenvalue/eigenvector solutions of large finite element models. The equations (1) to (4) of the March 30, 2013 Tech Brief are used as the component mode synthesis steps to obtain approximate frequencies and mode shapes of the dynamic system. However, these equations are also the first iteration of the Bathe subspace iteration method. Then iterating using
with obtained from the component mode synthesis solution, will lead to an accurate solution of the frequencies and mode shapes, see Chapter 11.6.2 of [1], and [2].
We illustrate this point and the possibilities provided in the solution scheme with two analyses.
In the first analysis, we consider the model of the front end of a tractor, seen in the movie above. This is a nonlinear analysis, in which the number of equations = 1,214,135, the number of contact equations = 3,546, and the number of frequencies required = 20. The number of iteration vectors = 40. First, we perform the component mode synthesis (CMS) solution with 10 static constraint mode vectors and 30 fixed interface vibration mode vectors. This corresponds to the first step of the subspace iteration. The required frequencies and mode shape vectors are approximate, in that the errors in the frequencies are about 10% on average. The predicted mode shapes are of course even less accurate.
Of course, in practice these errors are unknown. The errors have been established by also calculating the exact frequencies (to 6 digits) of the model, by simply continuing the subspace iterations, as given in Equations (1) to (5) above. The total number of subspace iterations needed for convergence is given in Table 1 which also lists experiences when more frequencies and mode shapes are sought.
Table 1 No. of subspace iterations for the front end tractor example; high precision results obtained
r = no. of static constraint modes
s = no. of fixed interface vibration modes
No. of required eigenvalues/ mode shapes p 
No. of subspace iterations including CMS step 
Standard subspace iteration (q = 2p) 
20  6 (r =10, s=30) 
6 
50  6 (r=10, s=90) 7 (r=20, s=80) 
15 
100  7 (r=10, s=190) 8 (r=40, s=160) 
18 
In the second example we consider the plasma fusion model, see Figure 1, also solved earlier, see also [2] below. Here we are now only interested in the lowest 20 frequencies and mode shapes, solved with 40 iteration vectors. This model is large because it contains about 5 million equations with about 700,000 Lagrange multiplier equations imposing contact conditions. The starting vectors can be established using a small or large number of static constraint modes, see Table 2. As seen in the table, the total solution time is not varying by very much. But this may of course be different in other analyses. The frequencies were solved on a single workstation with two quadcore processors running at 2.4 GHz and with a total of 64 GB of RAM.
Figure 1 Finite element model of a coil and support structure of a plasma fusion device
Table 2 Solution times for the plasma fusion model, CMS with
standard subspace iterations
r = no. of static constraint modes
s = no. of fixed interface vibration modes
No. of subspace iterations in the full iteration, including CMS step 
Time for calculating static constraint modes (sec) 
No. of subspace iterations to obtain fixed interface vibration modes 
Time for calculating fixed interface vibration modes (sec) 
Time for additional subspace iterations (sec) 
Total solution time (sec) 

Case 1 (r=10,s=30) 
10  567  10  1455  1305  3327 
Case 2 (r=20,s=20) 
10  670  9  1412  1285  3367 
Case 3 (r=30,s=10) 
15  733  8  1010  1496  3239 
Case 4 (r=40,s=0) 
18  791  0  0  1617  2408 
However, an advantage of the subspace iteration when used as a continuation of the CMS, is that there is no need to iterate to obtain a high precision answer, as required for the results referred to in Table 2. For example, performing another 2 iterations after the CMS step (using a total of 3 full subspace iterations) and also allowing for the fixed interface vibration modes only 4 iterations, reduces the solution time as shown in Table 3. In practice, the CMS step might be performed and thereafter just a few userspecified subspace iterations are used to improve the solution result, or to see by how much the solution result is changed, depending on the size of the problem solved and the available computational resources. We should note the good accuracy still obtained with fewer iterations, as shown in Figure 2.
Table 3 Solution times for the plasma fusion model, CMS with fewer subspace iterations
r = no. of static constraint modes
s = no. of fixed interface vibration modes
No. of subspace iterations in the full iteration, including CMS step 
Time for calculating static constraint modes (sec) 
No. of subspace iterations to obtain fixed interface vibration modes 
Time for calculating fixed interface vibration modes (sec) 
Time for addtional subspace iterations (sec) 
Total solution time (sec) 

Case 1 (r=10,s=30) 
3  567  4  618  809  1994 
Case 2 (r=20,s=20) 
3  670  4  660  786  2116 
Case 3 (r=30,s=10) 
3  733  4  632  827  2192 
Case 4 (r=40,s=0) 
3  791  0  0  778  1569 
Figure 2 First 20 modes of the plasma fusion model
While, clearly, experienced users of the CMS approach might get very accurate solutions to the frequencies and mode shapes sought (at a very reasonable cost) of a finite element model, in general the errors in the frequencies and mode shapes are unknown. However, the implementation in ADINA is that CMS can be followed by subspace iterations to give increasingly accurate solutions of frequencies and mode shapes, if required including nonlinear effects, like contact. The implementation of this scheme in ADINA provides users with powerful and flexible options for analyzing large and complex systems in industry.
References
 Bathe, K.J., Finite Element Procedures, Cambridge, MA: KlausJürgen Bathe, 2006.
 Bathe, K.J., The subspace iteration method  Revisited, Computers & Structures, 126:177183, 2013.
Keywords:
Frequencies, mode shapes, component mode synthesis, CraigBampton
scheme, Bathe subspace iteration method, fixed interface vibration modes, modes with contact conditions
^{*}The scheme is available in ADINA v. 9.