6 research outputs found
Adjoint methods for stellarator shape optimization and sensitivity analysis
Stellarators are a class of device for the magnetic confinement of plasmas without toroidal symmetry. As the confining magnetic field is produced by clever shaping of external electro-magnetic coils rather than through internal plasma currents, stellarators enjoy enhanced stability properties over their two-dimensional counterpart, the tokamak. However, the design of a stellarator with acceptable confinement properties requires numerical optimization of the magnetic field in the non-convex, high-dimensional spaces describing their geometry. Another major challenge facing the stellarator program is the sensitive dependence of confinement properties on electro-magnetic coil shapes, necessitating the construction of the coils under tight tolerances. In this Thesis, we address these challenges with the application of adjoint methods and shape sensitivity analysis.
Adjoint methods enable the efficient computation of the gradient of a function that depends on the solution to a system of equations, such as linear or nonlinear PDEs. Rather than perform a finite-difference step with respect to each parameter, one additional adjoint PDE is solved to compute the derivative with respect to any parameter. This enables gradient-based optimization in high-dimensional spaces and efficient sensitivity analysis. We present the first applications of adjoint methods for stellarator shape optimization.
The first example we discuss is the optimization of coil shapes based on the generalization of a continuous current potential model. We optimize the geometry of the coil-winding surface using an adjoint-based method, producing coil shapes that can be more easily constructed. Understanding the sensitivity of coil metrics to perturbations of the winding surface allows us to gain intuition about features of configurations that enable simpler coils. We next consider solutions of the drift-kinetic equation, a kinetic model for collisional transport in curved magnetic fields. An adjoint drift-kinetic equation is derived based on the self-adjointness property of the Fokker-Planck collision operator. This adjoint method allows us to understand the sensitivity of neoclassical quantities, such as the radial collisional transport and self-driven plasma current, to perturbations of the magnetic field strength. Finally, we consider functions that depend on solutions of the magneto-hydrodynamic (MHD) equilibrium equations. We generalize the well-known self-adjointness property of the MHD force operator to include perturbations of the rotational transform and the currents outside the confinement region. This self-adjointness property is applied to develop an adjoint method for computing the derivatives of such functions with respect to perturbations of coil shapes or the plasma boundary. We present a method of solution for the adjoint equations based on a variational principle used in MHD stability analysis
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 starâs 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 âbox in a starâ 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., |âšds/drâ©| â Ω^4/5.This research has been supported by the European Research Council, from the European Unionâs Horizon 2020 research and innovation programme, under grant agreement No. 337705 (CHASM), and by a Consolidated Grant from the UK STFC (ST/J001627/1)
On the relations between B2V Ms and RungeâKutta collocation methods
AbstractThe principal aim of this paper is a rigorous analysis of the relations between Block Boundary Value Methods (B2V Ms) with minimal blocksize defined over a suitable nonuniform finer mesh and well-known RungeâKutta collocation methods. Moreover, a further aspect that will be briefly investigated is the construction of an extended finer mesh for building B2V Ms with nonminimal blocksize. Some advantages that may arise from the use of the so-obtained methods will be also discussed
The hydrodynamics of ship sections entering and exiting a fluid
This thesis was submitted for the degree of Doctor of Philosophy and was awarded by Brunel University.We study the hydrodynamics of wedge,knuckle and box cross-sectional profiles undergoing transient extreme motions, in particular forced entry and exit at constant velocity or acceleration. Extensive data for the forces, pressures and free-surface profiles is generated by an extension of a fully-nonlinear boundary-integral method. The code is thoroughly checked by altering the time step and particle spacing on the bodies and Lagrangian free-surface markers, and, for the wedge,checking self-similarity for the infinite Froude number (gravity free) constant velocity entry. Difficulties with inviscid flow around sharp corners are discussed. Results for exit are of particular interest since no zero-gravity
approximation is valid and this precludes application of existing slamming theories in reverse. Whilst entry generally gives larger free-surface motions (spray jets), pressures and hence forces, calculation of exit is needed for the velocity of subsequent slamming and so is of practical interest too.
These results are compared with an approximate analytical model, based on Schwarz-Christoffel transformations to calculate the infinite-frequency added mass of the cross-section below the mean
water line. For constant acceleration of both entry and exit, the analytical theory is good during the early stages of motion. Later, the assumption of an undisturbed mean water level is clearly violated; the
exact calculations show a large amount of draw-down (up-rise), the free-surface making contact with the body well below (above) the mean water level. We therefore examine the effect of reducing (increasing) the submerged body volume to take account of this, which prolongs the agreement between
the results considerably and therefore might be used to improve practical calculation of extreme ship motions using existing strip theory codes.
Full sets of numerical data input/output are provided in the appendices, together with some mathematical details. We also speculate on the possible application of John's equation to wedge entry.Engineering and Physical Sciences Research Council (EPSRC) C.A.S.E. Award Schem
Annual Research Briefs, 1992
This report contains the 1992 annual progress reports of the Research Fellows and students of the Center for Turbulence Research. Considerable effort was focused on the large eddy simulation technique for computing turbulent flows. This increased activity has been inspired by the recent predictive successes of the dynamic subgrid scale modeling procedure which was introduced during the 1990 Summer Program. Several Research Fellows and students are presently engaged in both the development of subgrid scale models and their applications to complex flows. The first group of papers in this report contain the findings of these studies. They are followed by reports grouped in the general areas of modeling, turbulence physics, and turbulent reacting flows. The last contribution in this report outlines the progress made on the development of the CTR post-processing facility