## 4 Direct Time Integration Tailored to Systems with Friction Nonlinearities
### A Novel \"Piecewise-Linear\" Implementation for Direct Time Integration Methods  
Figure 4a clearly shows that, _since Newmark-\(\beta\) is an implicit scheme and the system at hand is nonlinear, an initial assumption_ on one of the unknown vectors at time \(\tau_{n+1}\), e.g., \({}^{n+1}\mathbf{q}\mathbf{p}\) is required. The first version of the damper model presented in [11] assumes \({}^{n+1}\mathbf{q}\mathbf{p}={}^{n}\mathbf{q}\mathbf{p}\). In other words, it assumes that the displacement keeps constant over a period \(\Delta\tau\). This assumption is not relied upon, as shown in Figure 4a and Appendix A.2: a predictor-corrector scheme (different from the one previously mentioned for the contact model) iterates until the assumed value is verified by the equilibrium equations. This assumption results in an average of \(\approx\)5 iterations per time step for frequencies \(<\)100 Hz and reaches as high as \(\approx\)11 for frequencies \(\geq\)200 Hz (see Figure 5a). Therefore, the impact on the computational time is quite negative and non-negligible. One may argue that better assumptions should be tried out, e.g., using the result of an explicit method as a starting guess. Many options have been tried out and the novel method proposed below was found to be the most suitable for our purpose: the number of iterations per time step is minimized thanks to an \"ad-hoc\" predictor-corrector.  
The novel initial assumption (i.e., predictor) is based on two strategic observations:  
* the contact state changes only a few times per period of vibration (4-7 in cases encountered in this work);
* if the contact state does not change from time step \(\tau_{n}\) to \(\tau_{n+1}\), the system can be considered perfectly linear (see Section 3)  
This novel formulation manages to derive a tentative value of all unknown vectors using the piecewise linearity of the contact model (Section 3), i.e., setting \(\mathbf{K}_{\mathbf{i}}(\tau_{n+1})=\mathbf{K_{i}}(\tau_{n})\ \forall i\). If the assumption is verified (i.e., the contact state is unchanged), as is the case in \(>\)90% of cases, only one iteration is required for each step. If it is not, then the procedure is repeated using, as an assumption, the updated \(\mathbf{K_{i}}\) matrices corresponding to the forecast contact state. Typically, only one additional iteration is required (see Figure 5b). The complete formulation is reported in Appendix A.2.  
In other words, the Newmark-\(\beta\) method is applied as if on a linear system for the entire duration of a given contact state (therefore, no iterations are required as described in [45]). Iterations are necessary only when a contact state transition occurs.  
The overall computational time is reduced by a factor of almost 50%, e.g., for a 100 Hz 3-period DTI run, the \"standard\" algorithm's the total time is 9.46 s, compared to the 5.09 s of the \"Piecewise linear\"  
Figure 5: Histogram representing the number of iterations per time step of the predictor-corrector scheme implemented in the DTI algorithm: (**a**) constant displacement assumption; (**b**) unchanged contact state assumption.  
implementation. Similar time savings have been found using different DTI integration schemes. Furthermore, it should be noted that this piecewise-linear implementation can be applied together with the adaptive time step to further increase time savings.