Several chemical engineering applications involve global optimization of process models in which nonlinear dynamics are encoded as ordinary differential equations (ODEs). In typical deterministic methods for global optimization, crucial lower bounds are generated by minimizing convex relaxations. Hence, we propose a new approach for generating convex relaxations of solutions of parametric ODEs, essentially by combining our recent optimization-based relaxation scheme for ODEs with the powerful Auxiliary Variable Method underlying the state-of-the-art solver BARON. We observe that the optimization problems defining each method can be combined, yielding tight, versatile ODE relaxations that are described by nonsmooth ODEs with convex optimal-value problems embedded.

The objective of this project is to explore the cardinality and algebraic properties of fluid dynamic models using a customized algorithm derived from the Rosenfeld-Gröbner algorithm. This analysis is designed to contrast classical deterministic models with the solutions obtained from an algebraic standpoint, offering a novel perspective that opens the scope in the search for potential solutions in fluid mechanics models.

Reduced dynamics obtained via the normal form style in the parametrisation method for invariant manifolds are analyzed for nonlinear vibratory systems. Thanks to symbolic and numerical asymptotic expansions, systematic developments allow one to show how the solutions are a generalization to arbitrary orders of the results obtained with perturbative approaches, in the more general context of efficient and direct reduction techniques. Illustrative examples on simple systems are shown, which can then be easily generalized to low-dimensional reduced dynamics obtained from large-scale models.

In this paper, the dynamics of pipes conveying pulsating fluid are investigated. The fluid flow velocity inside the simply supported pipe is assumed to be small and varying harmonically with a frequency Ω. The linear equations of motion are examined using multiple time-scales perturbation methods. The pipe structure is modelled as a beam-like, as a stretched beam-like and as a string-like problem. The respective partial differential equations governing the dynamics of the system are studied leading to infinite-dimensional systems of ordinary differential equations. When the fluid pulsation frequency Ω is equal to a natural frequency, the sum of two natural frequencies, or the difference of two natural frequencies, the pipe system is observed to exhibit complicated dynamical behaviour. Due to excitations of higher-order modes and internal resonances, it has been shown that applying the widely used Galerkin truncation method can lead to incorrect results.