Abstract: The potential for acute shortages of ventilators at the peak of the COVID-19 pandemic has raised the possibility of needing to support two patients from a single ventilator. To provide a system for understanding and prototyping designs we have developed a mathematical model of two patients supported by a mechanical ventilator. We propose a standard setup where we simulate the introduction of T-splitters to supply air to two patients and a modified setup where we introduce a variable resistance in each inhalation pathway and one-way valves in each exhalation pathway. Using the standard setup, we demonstrate that ventilating two patients with mismatched lung compliances from a single ventilator will lead to clinically-significant reductions in tidal volume in the patient with the lowest respiratory compliance. Using the modified setup, we demonstrate that it could be possible to achieve the same tidal volumes in two patients with mismatched lung compliances, and we show that the tidal volume of one patient can be manipulated independently of the other. The results indicate that, with appropriate modifications, two patients could be supported from a single ventilator with independent control of tidal volumes.

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Abstract: Nodal point sets, and associated collocation projections, play an important role in a range of high-order methods, including Flux Reconstruction (FR) schemes. Historically, efforts have focused on identifying nodal point sets that aim to minimise the L^∞ error of an associated interpolating polynomial. The present work combines a comprehensive review of known approximation theory results, with new results, and numerical experiments, to motivate that in fact point sets for FR should aim to minimise the L² error of an associated interpolating polynomial. New results include identification of a nodal point set that minimises the L² norm of an interpolating polynomial, and a proof of the equivalence between such an interpolating polynomial and an L² approximating polynomial with coefficients obtained using a Gauss-Legendre quadrature rule. Numerical experiments confirm that FR errors can be reduced by an order-of-magnitude by switching from popular point sets such as Chebyshev, Chebyshev-Lobatto and Legendre-Lobatto to Legendre point sets.

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Summary: A PhD Studentship is currently available. The project, will involve addition of 'in-situ' visualisation, processing, and analysis technology to PyFR, an open-source high-order massively-parallel CFD platform, as well as its application to solve a range of challenging unsteady flow problems. Candidates should hold, or expect to obtain, an undergraduate degree in a numerate discipline. Previous programming experience is important (ideally Python, C++ and CUDA).

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