AC
A.L.G. Cummins
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Mimetic Discretisation on Manifolds with Curvature
A Structure-Preserving Discretisation of Bundle-Valued Differential Forms on Riemannian Manifolds with Arbitrary Connection
Structure-preserving, or mimetic, discretisations of scalar-valued differential forms have been successfully applied to various fields in physics, engineering and computational modelling. However, scant attention has been paid to higher-order discretisations of differential forms with values in a vector bundle, despite the great advantages such discretisations would confer. A major obstacle to this endeavour is the fact that vectors on a general manifold are each defined in the tangent space to the manifold as a combination of local basis vectors. It is not immediately apparent how vectors on a curved manifold can be interpolated with scalar polynomials, as the local basis varies from point to point.
In this thesis, a mimetic discretization of bundle-valued forms in two dimensions is presented. We show that, for a 0-form, the exterior covariant derivative and the projector commute, and that the second exterior covariant derivative recovers the action of the curvature 2-form in the limit of mesh refinement. Covariant versions of the standard incidence and Hodge matrices are also introduced. These operators are used to compute the exterior covariant derivative and to solve the Hodge Laplacian on the unit 2-sphere, with spectral convergence being achieved in both cases.
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In this thesis, a mimetic discretization of bundle-valued forms in two dimensions is presented. We show that, for a 0-form, the exterior covariant derivative and the projector commute, and that the second exterior covariant derivative recovers the action of the curvature 2-form in the limit of mesh refinement. Covariant versions of the standard incidence and Hodge matrices are also introduced. These operators are used to compute the exterior covariant derivative and to solve the Hodge Laplacian on the unit 2-sphere, with spectral convergence being achieved in both cases.
...
Structure-preserving, or mimetic, discretisations of scalar-valued differential forms have been successfully applied to various fields in physics, engineering and computational modelling. However, scant attention has been paid to higher-order discretisations of differential forms with values in a vector bundle, despite the great advantages such discretisations would confer. A major obstacle to this endeavour is the fact that vectors on a general manifold are each defined in the tangent space to the manifold as a combination of local basis vectors. It is not immediately apparent how vectors on a curved manifold can be interpolated with scalar polynomials, as the local basis varies from point to point.
In this thesis, a mimetic discretization of bundle-valued forms in two dimensions is presented. We show that, for a 0-form, the exterior covariant derivative and the projector commute, and that the second exterior covariant derivative recovers the action of the curvature 2-form in the limit of mesh refinement. Covariant versions of the standard incidence and Hodge matrices are also introduced. These operators are used to compute the exterior covariant derivative and to solve the Hodge Laplacian on the unit 2-sphere, with spectral convergence being achieved in both cases.
In this thesis, a mimetic discretization of bundle-valued forms in two dimensions is presented. We show that, for a 0-form, the exterior covariant derivative and the projector commute, and that the second exterior covariant derivative recovers the action of the curvature 2-form in the limit of mesh refinement. Covariant versions of the standard incidence and Hodge matrices are also introduced. These operators are used to compute the exterior covariant derivative and to solve the Hodge Laplacian on the unit 2-sphere, with spectral convergence being achieved in both cases.
Electric island H2opper
A hydrogen powered electrical aircraft
Bachelor thesis
(2021)
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A.L.G. Cummins, C. Dolman, T.A. Gehlen, T.J.W. Koning, W.A. Mathoera, H.R.A.M. Spaan, D.I. Timmermans, Dick Daniel Vlot, O. van Voorst, S.G. Wulffraat, J.A. Pascoe, J. Lee
Aviation is a vital lifeline for island communities. However, air travel is also a significant source of air pollution and therefore contributes to global warming. The rising sea levels, caused by global warming, heavily affect the livability of pacific islands, causing drink water shortages and crop failure. Sustainable aviation has to be a part of the climate change solution.
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Aviation is a vital lifeline for island communities. However, air travel is also a significant source of air pollution and therefore contributes to global warming. The rising sea levels, caused by global warming, heavily affect the livability of pacific islands, causing drink water shortages and crop failure. Sustainable aviation has to be a part of the climate change solution.