12 Towards a Parallel Time Integration Method for Nonlinear Systems 143 uB3(m) time (s) 0 2 4 6 8 10 0 1 2 3 Fig. 12.5 Response of mass mB3; showing the reference solution ( ); and the one obtained with the decomposed time integration ( ) time (s) 0 2 4 6 8 10 error |qref−qsub| |qref| 10−8 10−10 10−12 10−14 10−16 Fig. 12.6 Global error of the decomposed time integration, as given in Eq. (12.27) where tol denotes the tolerance setting. Visually, no difference between the responses can be seen, but in order to show that the responses of both simulations are exactly identical, a global error measure has been determined, n D j u.ref / n u.sub/ n j ju.ref / n j ; (12.27) which is shown in Fig. 12.6. Here, one can clearly see that the decomposed time integration yields the exact same response. 12.4.3 Effect of Setting the Local and Global Tolerances In order to show the effect of the global and local tolerance settings on the number of iterations and computational time, we’ve defined a ratio between the two, ratio D LoTol GloTol : (12.28) By fixing the global tolerance to (10 8) and varying the ratio given in Eq. (12.28) between1and1010, one could get an idea of the optimal ratio between the local and global tolerances. The result of this analysis is shown in Fig. 12.7. Note that we cannot draw any conclusions for the general case about this, as this setting could very well be problem dependent as well. Nonetheless, some interesting effects are shown here. Firstly, for low ratios the problem is dominated by the local iterations, as a too small ratio will require more local iterations then strictly required. Also note that the number of global iterations are at its minimum here. One can also clearly identify the subsystem with the strongest nonlinearities, as subsystem A requires about 3 times the number of iterations that subsystem B requires. By increasing the ratio between the tolerances, the number of local iterations decrease for both substructures, while keeping the global iterations at a constant level. At a certain moment, the iterations of subsystem B and the global iterations start to “couple”, meaning that subsystem B only requires (approximately) one iteration per global iteration. In addition to this, it appears that Subsystem A also starts to converge towards its “iteration asymptote”. As can be seen from the lower graph, this point also denotes the minimum in terms of CPU time. By further increasing the ratio, the number of global iterations (and hence of subsystem B) start to increase, without any substantial decrease in the number of iterations for subsystem A, thereby also increasing the CPU time required. Finally, at extremely large ratios the method will degrade into the method shown in Fig. 12.1a and will require the same amount of iterations for subsystem A, B and the global interface problem, which are also equal to the number of iterations required for the reference solution.
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