An anisotropic mesh adaptation method for the finite element solution of heterogeneous anisotropic diffusion problems

Heterogeneous anisotropic diffusion problems arise in the various areas of science and engineering including plasma physics, petroleum engineering, and image processing. Standard numerical methods can produce spurious oscillations when they are used to solve those problems. A common approach to avoid this difficulty is to design a proper numerical scheme and/or a proper mesh so that the numerical solution validates the discrete counterpart (DMP) of the maximum principle satisfied by the continuous solution. A well known mesh condition for the DMP satisfaction by the linear finite element solution of isotropic diffusion problems is the non-obtuse angle condition that requires the dihedral angles of mesh elements to be non-obtuse. In this paper, a generalization of the condition, the so-called anisotropic non-obtuse angle condition, is developed for the finite element solution of heterogeneous anisotropic diffusion problems. The new condition is essentially the same as the existing one except that the dihedral angles are now measured in a metric depending on the diffusion matrix of the underlying problem. Several variants of the new condition are obtained. Based on one of them, two metric tensors for use in anisotropic mesh generation are developed to account for DMP satisfaction and the combination of DMP satisfaction and mesh adaptivity. Numerical examples are given to demonstrate the features of the linear finite element method for anisotropic meshes generated with the metric tensors.Comment: 34 page

Finite-difference schemes for anisotropic diffusion

In fusion plasmas diffusion tensors are extremely anisotropic due to the high temperature and large magnetic field strength. This causes diffusion, heat conduction, and viscous momentum loss, to effectively be aligned with the magnetic field lines. This alignment leads to different values for the respective diffusive coefficients in the magnetic field direction and in the perpendicular direction, to the extent that heat diffusion coefficients can be up to 10 to the 12 th times larger in the parallel direction than in the perpendicular direction. This anisotropy puts stringent requirements on the numerical methods used to approximate the MHD-equations since any misalignment of the grid may cause the perpendicular diffusion to be polluted by the numerical error in approximating the parallel diffusion. Currently the common approach is to apply magnetic field-aligned coordinates, an approach that automatically takes care of the directionality of the diffusive coefficients. This approach runs into problems at x-points and at points where there is magnetic re-connection, since this causes local non-alignment. It is therefore useful to consider numerical schemes that are tolerant to the misalignment of the grid with the magnetic field lines, both to improve existing methods and to help open the possibility of applying regular non-aligned grids. To investigate this, in this paper several discretisation schemes are developed and applied to the anisotropic heat diffusion equation on a non-aligned grid.</p

Stellar Models with Magnetism and Rotation: Mixing Length Theories and Convection Simulations

Some low-mass stars appear to have larger radii than predicted by standard 1D structure models; prior work has suggested that inefficient convective heat transport, due to rotation and/or magnetism, may ultimately be responsible. In this thesis, we explore this possibility using a combination of 1D stellar models, 2D and 3D simulations, and analytical theory. First, we examine this issue using 1D stellar models constructed using the Modules for Experiments in Stellar Astrophysics (MESA) code. We begin by considering standard models that do not explicitly include rotational/magnetic effects, with convective inhibition modelled by decreasing a depth-independent mixing length theory (MLT) parameter αMLT. We provide formulae linking changes in αMLT to changes in the interior specific entropy, and hence to the stellar radius. Next, we modify the MLT formulation in MESA to mimic explicitly the influence of rotation and magnetism, using formulations suggested by Stevenson (1979) and MacDonald and Mullan (2014) respectively. We find rapid rotation in these models has a negligible impact on stellar structure, primarily because a star’s adiabat, and hence its radius, is predominantly affected by layers near the surface; convection is rapid and largely uninfluenced by rotation there. Magnetic fields, if they influenced convective transport in the manner described by MacDonald and Mullan (2014), could lead to more noticeable radius inflation. Finally, we show that these non-standard effects on stellar structure can be fabricated using a depth-dependent αMLT: a non-magnetic, non-rotating model can be produced that is virtually indistinguishable from one that explicitly parameterises rotation and/or magnetism using the two formulations above. We provide formulae linking the radially-variable αMLT to these putative MLT reformulations. We make further comparisons between MLT and simulations of convection, to establish how heat transport and stellar structure are influenced by rotation and magnetism, by looking at the entropy content of 2D local and 3D global convective calculations. Using 2D “box in a star” simulations, created using the convection code Dedalus, we investigate changes in bulk properties of the specific entropy for increasingly stratified domains. We observe regions stable against convection near the bottom boundary, resulting in the specific entropy in the bulk of the domain exceeding the bottom boundary value: this could be a result of physical effects, such as increased amounts of viscous dissipation for more supercritical, highly stratified cases, but may also be influenced by the artificial boundary conditions imposed by these local simulations. We then turn to 3D global simulations, created using the convection code Rayleigh, and investigate these same properties as a function of rotation rate. We find the average of the shell-averaged specific entropy gradient in the middle third of the domain to scale with rotation rate in a similar fashion to the scaling law derived via MLT arguments in Barker et al. (2014), i.e., |⟨ds/dr⟩| ∝ Ω^4/5.This research has been supported by the European Research Council, from the European Union’s Horizon 2020 research and innovation programme, under grant agreement No. 337705 (CHASM), and by a Consolidated Grant from the UK STFC (ST/J001627/1)

The geodesic acoustic mode in strongly-shaped tight aspect ratio tokamaks

This thesis presents comparison between experimental measurements from the spherical tokamak MAST, two-fluid simulation data and theory of the Geodesic Acoustic Mode (GAM) in tight aspect ratio strongly shaped tokamak plasmas. The first identification of a strong ~10kHz mode detected in both potential and density fluctuations of the edge plasma in MAST using a reciprocating probe is given. The mode is radially localised, with outer limit ~ 2cm inside the separatrix, and is affected on application of resonant magnetic perturbations (RMP) generated by external coils. A shift in frequency with plasma rotation is found, and a suppression of the mode is observed above a certain threshold. Non-linear coupling to high wave number turbulence is evident, and an increase in power of turbulence fluctuations is seen after suppression. These observations are then interpreted in the context of known low frequency plasma modes present in the toroidal configuration. The supposition that the observed mode is a geodesic acoustic mode is considered and motivated by experimental observations and numerical simulations