Computational Magnetohydrodynamics Research Articles

Techniques, Tricks, and Algorithms for Efficient GPU-Based Processing of Higher Order Hyperbolic PDEs

GPU computing is expected to play an integral part in all modern Exascale supercomputers. It is also expected that higher order Godunov schemes will make up about a significant fraction of the application mix on such supercomputers. It is, therefore, very important to prepare the community of users of higher order schemes for hyperbolic PDEs for this emerging opportunity. We focus on three broad and high-impact areas where higher order Godunov schemes are used. The first area is computational fluid dynamics (CFD). The second is computational magnetohydrodynamics (MHD) which has an involution constraint that has to be mimetically preserved. The third is computational electrodynamics (CED) which has involution constraints and also extremely stiff source terms. Together, these three diverse uses of higher order Godunov methodology, cover many of the most important applications areas. In all three cases, we show that the optimal use of algorithms, techniques and tricks, along with the use of OpenACC, yields superlative speedups on GPUs! As a bonus, we find a most remarkable and desirable result: some higher order schemes, with their larger operations count per zone, show better speedup than lower order schemes on GPUs. In other words, the GPU is an optimal stratagem for overcoming the higher computational complexities of higher order schemes! Several avenues for future improvement have also been identified. A scalability study is presented for a real-world application using GPUs and comparable numbers of high-end multicore CPUs. It is found that GPUs offer a substantial performance benefit over comparable number of CPUs, especially when all the methods designed in this paper are used.

Code-to-Code Comparison for a PbLi Mixed-Convection MHD Flow

In “An Approach to Verification and Validation of MHD Codes for Fusion Applications” [S. Smolentsev et al., Fusion Eng. Des., Vol. 100, p. 65 (2015)], an effort for verification and validation of computer codes for liquid metal flows in a magnetic field for fusion cooling/breeding applications was initiated. The current study continues that effort. A group of experts in computational magnetohydrodynamics from several institutions in the United States and Europe performed a code-to-code comparison for the selected reference case of a mixed-convection buoyancy-opposed magnetohydrodynamic flow of eutectic lead-lithium (PbLi) alloy in a thin-wall conducting square duct at Hartmann number Ha = 220, Reynolds number Re = 3040, and Grashof number Gr = 2.88 × 107. As shown, the reference flow demonstrates a boundary layer separation in the heated region and formation of a reversed flow zone. The results of the comparison suggest that all five solvers predict well the key flow features but have moderate quantitative differences, in particular, in the location of the separation point. Also, two of the codes are more computationally dissipative, showing no velocity and temperature oscillations.

Magnetohydrodynamics modelling of a permanent magnets activated MRF clutch–brake

Chattering due to vibrations on the machine tool shaft and control of the transmitted torque are both a concern in precision machining. Hence, a magnetorheological fluid clutch–brake, also acting as a rotary damper could be a promising solution. Permanent magnets are used to mechanically activate the clutch thus offering simplicity and reliability in operation. The viscosity of the magnetorheological fluid changes with the action of the magnetic field thus allowing to tune the transmitted torque. Moreover, the fluid adds damping in the system. In this paper, a computational magnetohydrodynamics model has been realized in order to numerically investigate and visualize the torque transmission under different rotational velocities when submitted to magnetic flux. The model was validated with experimental data from the literature. Results showed that because the permanent magnets have a limited flux intensity, the torque transmission is only efficient at lower rotational velocities.

Wavelet-based parallel dynamic mesh adaptation for magnetohydrodynamics in the AMROC framework

Computational magnetohydrodynamics (MHD) for space physics has become an essential area in understanding the multiscale dynamics of geophysical and astrophysical plasma processes, partially motivated by the lack of space data. Full MHD simulations are typically very demanding and may require substantial computational efforts. In particular, computational space-weather forecasting is an essential long-term goal in this area, motivated for instance by the needs of modern satellite communication technology. We present a new feature of a recently developed compressible two- and three-dimensional MHD solver, which has been successfully implemented into the parallel AMROC (Adaptive Mesh Refinement in Object-oriented C++) framework with improvements concerning the mesh adaptation criteria based on wavelet techniques. The developments are related to computational efficiency while controlling the precision using dynamically adapted meshes in space-time in a fully parallel context.

Pulsed fusion space propulsion: Computational Magneto-Hydro Dynamics of a multi-coil parabolic reaction chamber

Pulsed fusion propulsion might finally revolutionise manned space exploration by providing an affordable and relatively fast access to interplanetary destinations. However, such systems are still in an early development phase and one of the key areas requiring further investigations is the operation of the magnetic nozzle, the device meant to exploit the fusion energy and generate thrust. One of the last pulsed fusion magnetic nozzle design is the so called multi-coil parabolic reaction chamber: the reaction is thereby ignited at the focus of an open parabolic chamber, enclosed by a series of coaxial superconducting coils that apply a magnetic field. The field, beside confining the reaction and preventing any contact between hot fusion plasma and chamber structure, is also meant to reflect the explosion and push plasma out of the rocket. Reflection is attained thanks to electric currents induced in conductive skin layers that cover each of the coils, the change of plasma axial momentum generates thrust in reaction. This working principle has yet to be extensively verified and computational Magneto-Hydro Dynamics (MHD) is a viable option to achieve that. This work is one of the first detailed ideal-MHD analysis of a multi-coil parabolic reaction chamber of this kind and has been completed employing PLUTO, a freely distributed computational code developed at the Physics Department of the University of Turin. The results are thus a preliminary verification of the chamber's performance. Nonetheless, plasma leakage through the chamber structure has been highlighted. Therefore, further investigations are required to validate the chamber design. Implementing a more accurate physical model (e.g. Hall-MHD or relativistic-MHD) is thus mandatory, and PLUTO shows the capabilities to achieve that.

Stationary axisymmetric configuration of the resistive thick accretion tori around a Schwarzschild black hole

We examine a thick accretion disc in the presence of external gravity and intrinsic dipolar magnetic field due to a non-rotating central object. In this paper, we generalize the Newtonian theory of stationary axisymmetric resistive tori of Tripathy, Prasanna $\&$ Das (1990) by including the fully general relativistic features. If we are to obtain the steady state configuration, we have to take into account the finite resistivity for the magnetofluid in order to avoid the piling up of the field lines anywhere in the accretion discs. The efficient value of conductivity must be much smaller than the classical conductivity to be astrophysically interesting. The accreting plasma in the presence of an external dipole magnetic field gives rise to a current in the azimuthal direction. The azimuthal current produced due to the motion of the magnetofluid modifies the magnetic field structure inside the disc and generates a poloidal magnetic field for the disc. The solutions we have found show that the radial inflow, pressure and density distributions are strongly modified by the electrical conductivity both in relativistic and Newtonian regimes. However, the range of conductivity coefficient is different for both regimes, as well as that of the angular momentum parameter and the radius of the innermost stable circular orbit. Furthermore, it is shown that the azimuthal velocity of the disc which is not dependent on conductivity is sub-Keplerian in all radial distances for both regimes. Owing to the presence of pressure gradient and magnetic forces. This work may also be important for the general relativistic computational magnetohydrodynamics that suffers from the lack of exact analytic solutions that are needed to test computer codes.

An analytical solution to electromagnetically coupled duct flow in MHD

The flow of an electrically conducting fluid in an array of square ducts, separated by arbitrary thickness conducting walls, subject to an applied magnetic field is studied. The analytical solution presented here is valid for thick walls and is based on the homogeneous solution obtained by Shercliff (Math. Proc. Camb. Phil. Soc., vol. 49 (01), 1953, pp. 136–144). Arrangements of ducts arise in a number of applications, most notably in fusion blankets, where liquid metal is used both as coolant and for tritium generation purposes. Analytical solutions, such as those presented here, provide insight into the physics and important benchmarking and validation data for computational magnetohydrodynamics (MHD), as well as providing approximate flow parameters for 1D systems codes. It is well known that arrays of such ducts with conducting walls exhibit varying degrees of coupling, significantly affecting the flow. An important practical example is the so-called Madarame problem (Madarame et al., Fusion Technol., vol. 8, 1985, pp. 264–269). In this work analytical results are derived for the relevant hydrodynamic and magnetic parameters for a single duct with thick walls analogous to the Hunt II case. These results are then extended to an array of such ducts stacked in the direction of the applied magnetic field. It is seen that there is a significant coupling affect, resulting in modifications to pressure drop and velocity profile. In certain circumstances, counter-current flow can occur as a result of the MHD effects, even to the point where the mean flow is reversed. Such phenomena are likely to have significant detrimental effects on both heat and mass transfer in fusion applications. The dependence of this coupling on parameters such as conductivities, wall thickness and Hartmann number is studied.

Computational magnetohydrodynamics in the simulation of gas circuit breakers

This paper summarises recent simulation approaches to compute the flow and field conditions in gas circuit breakers during the high current phase. The gas circuit breakers under consideration are low voltage circuit breakers that operate in air, as well as SF

Numerical calibration of a Lorentz force flowmeter

Numerical simulation of complex-shaped devices for contactless electromagnetic flow measurement in metallurgy is a challenge for computational magnetohydrodynamics. We report a series of numerical simulations which demonstrate for the first time that it is possible to predict the calibration constant of a generic Lorentz force flowmeter (LFF) with an uncertainty close to the requirements of real-life industrial applications. Our simulations involve both magnetostatic computations of a complex-shaped magnet system and magnetohydrodynamic computations of the flow of a liquid metal in a nozzle under the influence of a predominantly transverse magnetic field. In order to assess the role of turbulence, the simulations have been performed both for laminar and for turbulent flows using Reynolds-averaged Navier–Stokes equations in the latter case. In addition to the numerical simulations we have measured the calibration constant of the considered LFF using room-temperature liquid metal instead of liquid aluminum. A comparison between the numerically predicted and the measured values of the calibration constant shows that they differ by only 3.4%. This result suggests that numerical calibration of a LFF may become an economic alternative to expensive full-scale experimental calibration.

Theoretical Limitations of Discrete Exterior Calculus in the Context of Computational Electromagnetics

Discrete exterior calculus (DEC) attempts to mimic basic operations on differential forms in a discrete setting. This paper considers two concrete instances of how the completion of the DEC program is unachievable, and outlines the practical implications for computational electromagnetics. The two problems are the Commutative Cochain Problem (CCP) and the Discrete Star Localization Problem (DSLP). This paper elaborates on how the CCP is related to problems in computational magnetohydrodynamics, and how DSPL is related to material modeling and constructing discrete Hodge star operators whose matrix representations are maximally sparse in some concrete sense. A final section elaborates on the problem of Poincare duality on the level of cochains, and how the metric-free Whitney form finite element discretization of helicity functionals provides a model of how some of these theoretical obstacles can be side-stepped in three dimensions and, more generally, in dimensions.

A High-Temperature, Thermal Non-equilibrium Equation of State for Ammonia

The engineering applications of ammonia extend far beyond the pressure and temperature ranges for which thermodynamic models currently exist in the literature. Thus, a thermal non-equilibrium thermochemical model was developed to compute the composition and thermodynamic properties of ammonia for an extended temperature and pressure range that includes ionization reactions. Thermal non-equilibrium between electrons and heavy particles was included and is presented for ratios of 1/2, 1, 2 and 3. The fourteen-equation nonlinear system produced under the assumptions of ideal gas and two-temperature local thermodynamic equilibrium was solved numerically using a Newton-Raphson method. The thermochemical model is verified for both the composition and thermodynamic properties by comparisons to existing thermochemical models in the literature. These comparisons verify the model for the available, yet limited, temperature and density ranges. Analysis of the composition and thermodynamic properties as a function of the independent properties confirms the necessity for such a model as part of rigorous computations with computational fluid dynamics (CFD) or magnetohydrodynamics (MHD) computer codes. The model can be easily cast in tabular form to complement the set of conservation equations utilized by such codes.

Magnetized tori around Kerr black holes: analytic solutions with a toroidal magnetic field

The dynamics of accretion discs around galactic and extragalactic black holes may be influenced by their magnetic field. In this paper, we generalize the fully relativistic theory of stationary axisymmetric tori in Kerr metric of Abramowicz, Jaroszynski & Sikora by including strong toroidal magnetic field and construct analytic solutions for barotropic tori with constant angular momentum. This development is particularly important for the general relativistic computational magnetohydrodynamics that suffers from the lack of exact analytic solutions that are needed to test computer codes.

MHD Simulations of Crab's Jet and Torus

The Crab nebula is regarded as one of the most important “cosmic laboratories” in astrophysics, which has made a bigger impact on the development of astronomy than any other single object beyond the solar system. The most intriguing recent result is the completely unexpected discovery of a peculiar “jet-torus” structure in the inner part of the nebula. Similar structures were found later in other Crab-like nebulae. This discovery clearly indicates significant anisotropy of the wind from the Crab pulsar which has been ignored so far in simplified theoretical models of the nebula. Fortunately, the impressive progress in computational relativistic magnetohydrodynamics in recent years has made possible to study the Crab nebula without making such a drastic simplification of the problem. In this paper we present the results of the first study of such kind. They provide a likely explanation of the jet-torus pattern and show that the flow in the nebula is much more complex than it has been widely believed.

Divergence- and Curl-Preserving Prolongation and Restriction Formulas

We present new second-order prolongation and restriction formulas which preserve the divergence and, in some cases, the curl of a discretized vector field. The formulas are suitable for adaptive and hierarchical mesh algorithms with a factor-of-2 linear resolution change. We examine both staggered and collocated discretizations for the vector field on two- and three-dimensional Cartesian grids. The new formulas can be used in combination with numerical schemes that require a divergence-free solution in some discrete sense, such as the constrained transport schemes of computational magnetohydrodynamics. We also obtain divergence-preserving interpolation functions which may be used for streamline or field line tracing.

Computational Magnetohydrodynamics of Turbulence, Dynamos, and Jet Formation in Differentially Rotating Astrophysical Disks

We present the results of three-dimensional global magnetohydrodynamic (MHD) simulations of differentially rotating astrophysical disks. We simulate the time evolution of the disk by using a parallelized three-dimensional MHD code. Typical number of grid points is (Nr, Nφ, Nz) = (200, 64, 240) in a cylindrical coordinate system. We found that when the initial magnetic field is toroidal and weak (β = Pgas/Pmag 1), magnetic energy is amplified exponentially due to the dynamo action driven by the magnetorotational instability. In the nonlinear stage, magnetic turbulence excited in the disk tangles magnetic field lines. We found that the amplification of magnetic energy saturates when β ∼ 10 and that the system approaches a quasi-steady state. Inside the disk, filamentary shaped, magnetic pressure dominated (β < 1) regions appear. Magnetic energy release in low-β regions leads to violent time variations of X-ray emission from the disk. When the initial magnetic field is poloidal, magnetically driven collimated jet emanates from the surface of the disk.

A boundary condition for computational magnetohydrodynamics

A new “background plasma” method for treating boundaries in Eulerian magnetohydrodynamic computations is presented. Previously unconsidered aspects of plasma confinement by a wall are examined in the formulation of the boundary condition. Innovations include a distinct condition under which a plasma can separate from a wall resulting in an influx of “background plasma,” the use of boundary values which are interpreted to be interface values not included in volume integrations, and a condition under which a plasma returns to the wall resulting in a stoppage of the outflux of “background plasma.” Excellent agreement is obtained between a Lagrangian computation and an Eulerian computation incorporating the new boundary condition.

Conservation form in computational magnetohydrodynamics

The question of whether or not to use a conservation form of the internal energy equation in computational magnetohydrodynamics is considered. Conservation errors resulting from both spatial and temporal difference methods are demonstrated, and means of eliminating the spatial errors and reducing the temporal ones are discussed. It is shown that only in the limit dt -> 0 can a nonconservation form of the internal energy equation be used without destroying in the finite difference equations any of the conservation properties of the magnetohydrodynamic partial differential equations.