An iterative method for the approximation of fibers in slow-fast systems

In this paper we extend a method for iteratively improving slow manifolds so that it also can be used to approximate the fiber directions. The extended method is applied to general finite dimensional real analytic systems where we obtain exponential estimates of the tangent spaces to the fibers. The method is demonstrated on the Michaelis-Menten-Henri model and the Lindemann mechanism. The latter example also serves to demonstrate the method on a slow-fast system in non-standard slow-fast form. Finally, we extend the method further so that it also approximates the curvature of the fibers.Comment: To appear in SIAD

Canards in stiction: on solutions of a friction oscillator by regularization

We study the solutions of a friction oscillator subject to stiction. This discontinuous model is non-Filippov, and the concept of Filippov solution cannot be used. Furthermore some Carath\'eodory solutions are unphysical. Therefore we introduce the concept of stiction solutions: these are the Carath\'eodory solutions that are physically relevant, i.e. the ones that follow the stiction law. However, we find that some of the stiction solutions are forward non-unique in subregions of the slip onset. We call these solutions singular, in contrast to the regular stiction solutions that are forward unique. In order to further the understanding of the non-unique dynamics, we introduce a regularization of the model. This gives a singularly perturbed problem that captures the main features of the original discontinuous problem. We identify a repelling slow manifold that separates the forward slipping to forward sticking solutions, leading to a high sensitivity to the initial conditions. On this slow manifold we find canard trajectories, that have the physical interpretation of delaying the slip onset. We show with numerics that the regularized problem has a family of periodic orbits interacting with the canards. We observe that this family has a saddle stability and that it connects, in the rigid body limit, the two regular, slip-stick branches of the discontinuous problem, that were otherwise disconnected.Comment: Submitted to: SIADS. 28 pages, 12 figure

Integrated Digital Reconstruction of Welded Components: Supporting Improved Fatigue Life Prediction

In the design of offshore jacket foundations, fatigue life is crucial. Post-weld treatment has been proposed to enhance the fatigue performance of welded joints, where particularly high-frequency mechanical impact (HFMI) treatment has been shown to improve fatigue performance significantly. Automated HFMI treatment has improved quality assurance and can lead to cost-effective design when combined with accurate fatigue life prediction. However, the finite element method (FEM), commonly used for predicting fatigue life in complex or multi-axial joints, relies on a basic CAD depiction of the weld, failing to consider the actual weld geometry and defects. Including the actual weld geometry in the FE model improves fatigue life prediction and possible crack location prediction but requires a digital reconstruction of the weld. Current digital reconstruction methods are time-consuming or require specialised scanning equipment and potential component relocation. The proposed framework instead uses an industrial manipulator combined with a line scanner to integrate digital reconstruction as part of the automated HFMI treatment setup. This approach applies standard image processing, simple filtering techniques, and non-linear optimisation for aligning and merging overlapping scans. A screened Poisson surface reconstruction finalises the 3D model to create a meshed surface. The outcome is a generic, cost-effective, flexible, and rapid method that enables generic digital reconstruction of welded parts, aiding in component design, overall quality assurance, and documentation of the HFMI treatment.Comment: 6 pages, 7 figures, submitted to 2023 IEEE International Conference on Imaging Systems and Techniques (IST2023