MS03.7: Computational Methods

This work presents a novel approach that overcomes the sequential nature of numerical path continuation. The idea is to first define an appropriately simplified low-fidelity model and predict its relevant dynamics with low computational effort, e.g. analytically. Next, the set of relevant solution points on the target (high-fidelity) curve is computed using the low-fidelity solutions as departure. The proposed generic concept is exemplified for a selection of nonlinear vibration problems. Different types of system models, nonlinearities and analyses are considered, and the Harmonic Balance method is used in all cases to compute periodic limit states. In particular, it is shown that the proposed concept is applicable to modal and harmonic order refinement. Finally, it is shown that the concept is also interesting for system parameter sensitivity analyses and it permits to robustly reach parameter ranges that are extremely difficult to obtain with conventional path continuation.

We develop a fast data assimilation method for estimating the state of a dynamical system from its partial time series observations. Our method relies on discrete empirical interpolation method (DEIM) and therefore we refer to it as extended DEIM. Extended DEIM uses an auxiliary differential equation to approximate the optimal kernel vector which appears in DEIM. The dimension of the auxiliary equation is much smaller than the dimension of the original dynamical system. Therefore, our method is particularly attractive for data assimilation of high-dimensional systems. Furthermore, extended DEIM is intentionally designed to estimate the state of the system even when few sensor measurements are available.

We develop a new explicit integration method for transient dynamics computation of viscoelastic materials, surpassing the stability limits of Belytschko’s widely used half-lagged velocity approximation. Based on the central difference (CD) scheme, our method integrates the viscous stress strain law, while keeping an explicit scheme. Moreover, we provide a new stability analysis of the variable time step central difference method. We prove that CD’s stability can be ensured only thanks to the zero-stability criteria, based on the multistep formulation of the method. We perform analytical developments on a single degree of freedom problem. Thus, we show that, without viscous damping, ensuring stability requires to decrease, or at least to maintain constant the time step during the time integration. Furthermore, we prove that, with viscous damping, there exists a slight possibility of increasing the time step during the time integration.

We provide a bifurcation analysis of the coiling patterns in liquid threads falling onto a moving surface. In the inertialess regime (low fall heights), the geometric model of coiling predicts four primary coiling patterns. We report on the existence of additional stable patterns that occur at the same range of belt velocity as the meandering pattern, with more intricate patterns and longer periods.