Convergence and computational cost

CBSM solution convergence

In terms of how a suitable number of realisations was selected to build the correlation datasets of Figure 3.13 and Figure 3.17, 2000 instances were initially run for both the full and condensed model stochastic simulations. This exact choice was influenced by a combination of factors. Firstly, a sample of this size is considered to be at the high end of the spectrum for direct Monte Carlo simulations when only the first two statistical moments are of interest. Further details have been supplied in Section 2.3.2. Secondly, even samples containing as few as 100 stochastic instances have been shown to yield tractable results in practice, for example in the original CBSM articles. This was the case for tests ranging from a 1-D rod to a real satellite model. The number and type of random variables present did not appear to affect this result.

As a conservative scenario, the improved CBSM case employed ±50% NF randomisa- tion with uniform distribution, expected to show poor convergence due to the extreme perturbation limits. The standard definition of the physical space MCS, as per Sec- tion 3.2.2, was used. The average solution relative error for µ and σ was evaluated over all DOFs in terms of the number of completed solutions, for the full solution frequency range. The procedure pertains to the same principles as the construction of rµ and rσ.

The main findings can be seen on Figure 3.16.

0 50 100 150 200 250 300 350 400 Number of realisations 10-3 10-2 10-1 100 Relative error MCS, CBSM, , boundary DOFs CBSM, , modal DOFs MCS, mean

CBSM, mean, boundary DOFs CBSM, mean, modal DOFs

Figure 3.16: Maximum relative error of the CBSM and MCS with respect to solution based on 2000 MCS realisations. Convergence criterion set to 1% relative error. A convergence criterion of 1% maximum relative error was deemed a reasonable target. The condition was met at approximately 300 realisations for the mean and 400 for the standard deviation. From the observed trends, it is additionally apparent that the CBSM’s convergence rate is very similar to that of the MCS, regardless of whether degrees of freedom in physical or modal space representations are being compared. On the basis of this study, 500 realisations were considered as a reliable choice for executing CBSM-based analyses, providing some margin of safety over the number necessitated by the already conservatively defined test case.

Computational requirements against PSA

It is worth briefly discussing computational time of the CBSM against the baseline parametric MCS. The former would be expected to hold an advantage, in view of the sizeable reduction in the problem size owing to the CMS representation of the model. To quantify the performance of the method, two metrics were extracted from the executed simulations, with results summarised in Table 3.2. Solution time was comprised of various tasks, such as constructing or reassembling the mass and stiffness matrices, solving the GEP and recovering the required final outputs by modal superposition. Overhead was attributed to input/output (I/O) operations, the code facilitating the PSA and the original/improved CBSM implementations, as well as other activities unrelated to the principal computations.

Table 3.2: Computational time per realisation, 100 frequency points CBSM Implementation PSA Nastran Original Nastran Originala Matlab Improvedb Matlab Solution 297 s 11.9 s 12.5 s 0.6 s Overhead 436 s 4.7 s 0.4 s <0.1 s Total 733 s 16.6 s 12.9 s 0.7 s a

A custom code avoiding excessive I/O

b_{GEP reduction to an ordinary eigenvalue problem, no explicit reassembly}

The results affirm that in reality, the replacement of a PSA-based stochastic analysis with more efficient approaches can be easily justified from a speed of execution stand- point. The original CBSM accelerated the solution for 500 realisations from almost 102 h to about 2 h 18 min and 1 h 47 min, respectively, for two different deployments. The latter was further reduced by a factor of 18.4 with the improved CBSM, in other words when the transformation of the GEP to an ordinary eigenvalue problem is done. Its solution cost was dominated by solving the latter via a generic non-sparse algo- rithm, and the lack of explicit reassembly at each realisation significantly lessened the performance loss incurred due to excessive I/O operations.

It must be pointed out that the parametric MCS and the first rendition of the original CBSM were carried out in Nastran, while the second one and the improved CBSM were implemented entirely in Matlab, using mass and stiffness system matrices initially con- structed in Nastran. Therefore the stated overhead times are not necessarily analogous, as different forms of I/O operations were facilitated.

In terms of the fundamental solution times, all methods call highly optimised sub- routines using multi-core parallelism. Indeed, recreating Nastran’s Lanczos iterative solution in Matlab resulted in an average runtime of 12.5 s for the original CBSM with a similar number of calculated MVPs. This signifies the inadequacy of sparse iterative solvers to cope with relatively dense matrices, such as the ones occurring in CMS. Fi- nally, observe that the reduced SSTL300 satellite problem is quite small at 1569 DOFs. It is expected that for larger condensed systems, a gain in efficiency of the same order would be exhibited by the improved CBSM, since the algorithmic complexity remains identical, i.e. a cubic scaling.

Taking into account the promising results demonstrated on Figure 3.15(c) for solution recovery of non-boundary DOFs, it is also possible to assert that the nodes whose physical responses are required do not have to be identified prior to performing the model condensation, therefore do not need to be ascribed to the B set. This also enables the efficient storage of the complete numerical solution as a combination of physical and modal responses along with a set of transformation matrices. For the particular SSTL300 spacecraft case investigated here, the storage requirements are presented in Table 3.3. Note that this capability enabled the conduction of the parametric survey presented in Section 3.2.3, which consolidated data acquired from tens of thousands of CBSM executions.

Table 3.3: Solution vector storage requirements for a full stochastic simulation

Per realisationa Transformation 500 realisations

CBSM 2.4 MB 1382.3 MB 2.52 GB

PSA 371.9 MB 181.6 GB

a

For all DOFs over 100 frequency points

Generally, the Craig-Bampton stochastic method would allow saving the full results dataset in a sufficiently compact form even for much larger FE models and at more fre- quency points. This advantage further supplements the drastically reduced computing cost, especially pronounced for the improved CBSM variants.