Three-dimensional Cartesian Geometry Research Articles

An analytical nodal method for solving the SN transport equation in 2-D Cartesian geometry

A novel Analytical Nodal Discrete Ordinates (ANDO) method for the solution of the discrete ordinates (SN) neutron transport equation in Cartesian geometry with isotropic scattering and constant neutron source is presented. A nodal method approximates the multi-dimensional transport equation as a system of coupled one-dimensional transport equations along each coordinate axis through transverse integration. The resulting transverse-integrated equations are then solved separately using a one-dimensional solution technique. The one-dimensional solution utilized in the new ANDO method is based on a recent closed-form analytical solution in slab geometry (WANG and BYAMBAAKHUU, 2017). The novel solution approach allows for a compact closed-form solution on each computational cell. Results for constant and linear transverse leakage approximations are presented. The ANDO method possesses a number of favorable properties, demonstrated with numerical results, such as high accuracy, rapid convergence, asymptotic preserving, positivity preserving, near linear computational complexity, and local-hp adaptivity. The ANDO method can also be easily extended to higher-order transverse leakage approximations, to three-dimensional Cartesian geometry, and to multi-group.

Convection with misaligned gravity and rotation: simulations and rotating mixing length theory

ABSTRACT We present numerical simulations, using two complementary set-ups, of rotating Boussinesq thermal convection in a three-dimensional Cartesian geometry with misaligned gravity and rotation vectors. This model represents a small region at a non-polar latitude in the convection zone of a star or planet. We investigate the effects of rotation on the bulk properties of convection at different latitudes, focusing on determining the relation between the heat flux and temperature gradient. We show that our results may be interpreted using rotating mixing length theory (RMLT). The simplest version of RMLT (due to Stevenson) considers the single mode that transports the most heat. This works reasonably well in explaining our results, but there is a systematic departure from these predictions (up to approximately $30{{\ \rm per\ cent}}$ in the temperature gradient) at mid-latitudes. We develop a more detailed treatment of RMLT that includes the transport afforded by multiple modes, and we show that this accounts for most of the systematic differences. We also show that convectively generated zonal flows and meridional circulations are produced in our simulations, and that their properties depend strongly on the dimensions of the box. These flows also affect the heat transport, contributing to departures from RMLT at some latitudes. However, we find the theoretical predictions of the multi-mode theory for the mid-layer temperature gradient, the root-mean-square (rms) vertical velocity, the rms temperature fluctuation, and the spatial spectrum of the heat transport at different latitudes are all in reasonably good agreement with our numerical results when zonal flows are small.

A symmetry preserving scheme for three-dimensional Lagrangian radiation hydrodynamic simulations of ICF capsule implosion

A symmetry preserving scheme for three-dimensional (3D) Lagrangian radiation hydrodynamic simulations of an inertial confinement fusion (ICF) capsule implosion is presented. Violation of spherical symmetry exists in conventional scheme in 3D Cartesian coordinate system. Following the idea of Caramana et al. (E.J. Caramana, et al., A Compatible, Energy and Symmetry Preserving Lagrangian Hydrodynamics Algorithm in Three-Dimensional Cartesian Geometry, Journal of Computational Physics, 157 (2000) 89–119), a symmetry preserving scheme in 3D Cartesian coordinate system is extended to Lagrangian radiation hydrodynamics. This scheme is specially designed for spherical (radial) grids. Appropriate calculations of auxiliary points in the discretization of Lagrangian hydrodynamic equations are given. Small modification is applied to the flux-related Jacobian matrix in the diffusion scheme. The modified diffusion scheme can guarantee the symmetrical flux. The symmetry preserving properties of the developed scheme for Lagrangian radiation hydrodynamics are proved in theory. The developed symmetry preserving scheme is applied to the 3D simulations of an ICF capsule implosion. Numerical results show that the symmetry preserving scheme for Lagrangian radiation hydrodynamics helps to remove unphysical instability in the 3D simulations of ICF capsule implosion.

Numerical Simulations of Transverse Oscillations of a Finely Structured Solar Flux Tube

Abstract In the light of recent CRISP/SST observations we aim to study transverse oscillations of a finely structured flux tube. With the use of PLUTO code, we numerically solve ideal magnetohydrodynamic equations in three-dimensional Cartesian geometry. With a pressure signal we generate a finely structured magnetic flux tube that consists of a central strand and four off-central strands. This flux tube is perturbed by a periodic driver in the azimuthal component of velocity. This driver results in transverse oscillations that propagate upward along these strands. The central strand experiences torsional Alfvén waves, while the off-central strands exhibit fast magnetoacoustic kink oscillations. The latter are guided by the off-central strands, which are regions of depression in Alfvén speed.

Comparison of two three-dimensional heterogeneous variational nodal methods for PWR control rod cusping effect and pin-by-pin calculation

Two heterogeneous nodal methods based on the Variational Nodal Method (VNM) are investigated with diffusion approximation in three-dimensional Cartesian geometry. The first one is named as Function Expansion (FE) method while the second is Finite Sub-element (FS) method. Based on our previous work and the code Violet-Het1D in one-dimensional slab geometry, a code named Violet-Het3D was developed to handle the Pressurized Water Reactor (PWR) control rod cusping effect and pin-by-pin calculation by using either of these two methods. To eliminate the control rod cusping effect, Violet-Het3D provides a different idea from the existing methods. Neither homogenization procedure nor mesh adjustment is needed in Violet-Het3D by taking advantage of the treatment for heterogeneous nodes. Numerical results show that both the FE and FS methods can eliminate the cusping effect and obtain accurate power distribution while the FE method has relatively higher efficiency and accuracy. In contrast, for pin-by-pin calculation, the FS method obtains more accurate eigenvalue and pin power distribution than the FE method.

Improved discrete nodal transport method for treating void regions

To solve neutron transport problems with void regions accurately and efficiently, the legacy discrete nodal transport method (DNTM) was improved in three-dimensional Cartesian geometry. Firstly, the efficient nodal-equivalent finite difference (NEFD) algorithm was modified, named M-NEFD, to directly treat the zero total cross section in the denominator for void nodes. Furthermore, an angle projection discrete-ordinates (APSN) method was proposed to combine with the NEFD algorithm, named APSN-NEFD, to treat the void regions without sweeping the void nodes one by one. Thirdly, the angular sweeps within one octant of both M-NEFD and APSN-NEFD were parallelized using OpenMP. Based on these improvements, a discrete-ordinates nodal transport code named NECP-HONESTY has been developed in three-dimensional Cartesian geometry. To test the performance of the method and the code, a fission-source problem with two small void regions and a fixed-source problem with one large void region were presented in this paper by comparing with the Monte Carlo method. It has been demonstrated by the numerical results that the improved DNTM can provide accurate eigenvalue and scalar flux, no matter directly sweeping the void nodes with M-NEFD or skipping the void regions with APSN. In addition, compared with M-NEFD, APSN-NEFD can efficiently reduce both computing time and storage requirement for problem with large void regions.

A fully implicit numerical integration of the relativistic particle equation of motion

Relativistic strongly magnetized plasmas are produced in laboratories thanks to state-of-the-art laser technology but can naturally be found around compact objects such as neutron stars and black holes. Detailed studies of the behaviour of relativistic plasmas require accurate computations able to catch the full spatial and temporal dynamics of the system. Numerical simulations of ultra-relativistic plasmas face severe restrictions due to limitations in the maximum possible Lorentz factors that current algorithms can reproduce to good accuracy. In order to circumvent this flaw and repel the limit to$\unicode[STIX]{x1D6FE}\approx 10^{9}$, we design a new fully implicit scheme to solve the relativistic particle equation of motion in an external electromagnetic field using a three-dimensional Cartesian geometry. We show some examples of numerical integrations in constant electromagnetic fields to prove the efficiency of our algorithm. The code is also able to follow the electric drift motion for high Lorentz factors. In the most general case of spatially and temporally varying electromagnetic fields, the code performs extremely well, as shown by comparison with exact analytical solutions for the relativistic electrostatic Kepler problem as well as for linearly and circularly polarized plane waves.

Microwave-heating laboratory experiments for planetary mantle convection

Thermal evolution of telluric planets is mainly controlled by secular cooling and internal heating due to the decay of radioactive isotopes, two processes that are equivalent from the standpoint of convection dynamics. In a fluid cooled from above and volumetrically heated, convection is dominated by instabilities of the top boundary layer and the interior thermal structure is non-isentropic. Here we present innovative laboratory experiments where microwave radiation is used to generate uniform internal heat in fluids at high Prandtl number (${>}300$) and high Rayleigh–Roberts number (ranging from $10^{4}$ to $10^{7}$), appropriate for planetary mantle convection. Non-invasive techniques are employed to determine both temperature and velocity fields. We successfully validate the experimental results by conducting numerical simulations in three-dimensional Cartesian geometry that reproduce the experimental conditions. Scaling laws relating key characteristics of the thermal boundary layer, namely its thickness and temperature drop, to the Rayleigh–Roberts number have been established for both rigid and free-slip boundary conditions. A robust conclusion is that for rigid boundary conditions the internal temperature is significantly higher than for free-slip boundary conditions. Our scaling laws, coupled with plausible physical parameters entering the Rayleigh–Roberts number, enable us to calculate the mantle potential temperature for the Earth and Venus, two telluric planets with different mechanical boundary conditions at their surface.

Mantle plume characteristics in three-dimensional depth- and temperature-dependent models

Numerical calculations in three-dimensional Cartesian geometry have been carried out. The aim of the paper is to investigate systematically the effect of Rayleigh number and temperature- and depth-dependent viscosity on the radius and temperature anomaly of mantle plumes and on surface observable characteristics like topography and geoid anomaly. As reference models depth-dependent models were chosen, then the addition of weak and stronger temperature-dependence was studied. The Rayleigh number changed between \(10^5\) and \(10^7\). If viscosity decreases with temperature the convection becomes more vigorous and thermally weaker plumes rise from the bottom boundary layer. The radius and temperature anomaly of the upwelling decreases with increasing temperature-dependence. Topographic uplift significantly decreases with increasing temperature-dependence. For weak depth-dependent models the geoid decreases together with the topography as Rayleigh number increases. However, strongly depth-depend models show negative geod anomaly above the hot plume. In case of temperature- and depth dependent models the geoid have a maximum in the function of Rayleigh number. This maximum appears at different Rayleigh number depending on the actual factors of temperature- and depth-dependence. The amplitudes of topographic uplift in the case of stronger temperature-dependence and high Rayleigh number are not far from the observed values, but the geoid signal is higher by a factor of about 2.

Eulerian rotations of deformed nuclei for TDDFT calculations

We discuss three practical methods for performing Eulerian rotations of Slater determinants in a three-dimensional Cartesian geometry. In addition to the straightforward application of the active form of the quantum mechanical rotation operator, we introduce two methods using a passive position-space rotation followed by an active spin-space rotation, one after variation and the other before variation. These methods can be used to initialize reactions involving deformed nuclei where a particular alignment of the deformed nuclei with respect to the collision axis is desired. We show that doing the rotation before the variation is the most efficient way of generating such initial states.

One class of analytical self-similar solutions to the system of radiative transfer and energy equations for a homogeneous medium

The known class of analytical self-similar solutions of the traveling heat wave type for a nonlinear integro-differential system of equations which describes nonstationary spectral transfer of radiant energy in a kinetic model is generalized to a homogeneous medium. The solutions are constructed in three-dimensional Cartesian geometry with specially chosen absorption and scattering coefficients. No additional terms that might distort the essence of the physical phenomena being described are introduced into the original equations, and the solution is almost completely determined by the functional dependence on a self-similar variable. This class of exact solutions is also applicable to the “gray matter” approximation and can be used to test numerical methods for calculating direct and inverse problems of radiative transfer. Examples of test calculations are given.

Titan impacts and escape

We report on hydrodynamic calculations of impacts of large (multi-kilometer) objects on Saturn’s moon Titan. We assess escape from Titan, and evaluate the hypothesis that escaping ejecta blackened the leading hemisphere of Iapetus and peppered the surface of Hyperion. We carried out two- and three-dimensional simulations of impactors ranging in size from 4 to 100 km diameter, impact velocities between 7 and 15 km s −1, and impact angles from 0° to 75° from the vertical. We used the ZEUSMP2 hydrocode for the calculations. Simulations were made using three different geometries: three-dimensional Cartesian, two-dimensional axisymmetric spherical polar, and two-dimensional plane polar. Three-dimensional Cartesian geometry calculations were carried out over a limited domain (e.g. 240 km on a side for an impactor of size d i = 10 km), and the results compared to ones with the same parameters done by Artemieva and Lunine (2005); in general the comparison was good. Being computationally less demanding, two-dimensional calculations were possible for much larger domains, covering global regions of the satellite (from 800 km below Titan’s surface to the exobase altitude 1700 km above the surface). Axisymmetric spherical polar calculations were carried out for vertical impacts. Two-dimensional plane-polar geometry calculations were made for both vertical and oblique impacts. In general, calculations among all three geometries gave consistent results. Our basic result is that the amount of escaping material is less than or approximately equal to the impactor mass even for the most favorable cases. Amounts of escaping material scaled most strongly as a function of velocity, with high-velocity impacts generating the largest amount, as expected. Dependence of the relative amount of escaping mass f esc = m esc / M i on impactor diameter d i was weak. Oblique impacts (impact angle θ i > 45°) were more effective than vertical or near-vertical impacts; ratios of m esc / M i ∼ 1–2 were found in the simulations.

LOW MACH NUMBER MODELING OF TYPE IA SUPERNOVAE. IV. WHITE DWARF CONVECTION

We present the first three-dimensional, full-star simulations of convection in a white dwarf preceding a Type Ia supernova, specifically the last few hours before ignition. For these long-time calculations we use our low Mach number hydrodynamics code, MAESTRO, which we have further developed to treat spherical stars centered in a three-dimensional Cartesian geometry. The main change required is a procedure to map the one-dimensional radial base state to and from the Cartesian grid. Our models recover the dipole structure of the flow seen in previous calculations, but our long-time integration shows that the orientation of the dipole changes with time. Furthermore, we show the development of gravity waves in the outer, stable portion of the star. Finally, we evolve several calculations to the point of ignition and discuss the range of ignition radii.

An enhanced HOWFARLESS option for DOSXYZnrc simulations of slab geometries

The Monte Carlo code DOSXYZnrc is a valuable instrument for calculating absorbed dose within a three-dimensional Cartesian geometry. DOSXYZnrc includes several variance reduction techniques used to increase the efficiency of the Monte Carlo calculation. One such technique is HOWFARLESS which is used to increase the efficiency of beam commissioning calculations in homogeneous phantoms. The authors present an enhanced version of HOWFARLESS which extends the application to include phantoms inhomogeneous in one dimension. When the enhanced HOWFARLESS was used, efficiency increases as high as 14 times were observed without any loss in dose accuracy. The efficiency gains of an enhanced HOWFARLESS simulation was found to be dependent on both slab geometry and slab density. As the number of two-dimensional voxel layers per slab increases, so does the efficiency gain. Also, as the mass density of a slab is decreased, the efficiency gains increase.

Analytical and geometrical tools for 3D volume of fluid methods in general grids

It is well known that volume of fluid (VOF) methods in three-dimensions, especially those based on unsplit advection schemes, involve highly complex geometrical operations. The objective of this work is to propose, for general grids and three-dimensional Cartesian geometry, simple and efficient geometrical tools for volume truncation operations that typically arise in VOF methods and an analytical method for local volume enforcement. The results obtained for different tests and grid types show that the proposed analytical method may be as much as three times faster than Brent’s iterative method. Advection tests were carried out using hexahedral grids obtained from deformation of a cubic grid to assess the accuracy of the proposed tools in combination with a recently proposed unsplit PLIC–VOF method.

Spatial Finite-Element Lumping Techniques for the Quadrilateral Mesh Sn Equations in X-Y Geometry

We consider two general finite-element lumping techniques for the Sn equations with discontinuous finite-element spatial discretization and apply them to quadrilateral meshes in x-y geometry. One technique is designed to ensure a conservative approximation and is referred to as conservation preserving (CP). The other technique is designed to preserve the exact solution whenever it is contained within the trial space and is referred to as solution preserving (SP). These techniques are applied in x-y geometry on structured nonorthogonal grids using the bilinear-discontinuous finite-element approximation. The schemes are both theoretically analyzed and computationally tested. Analysis shows that the two lumping schemes are equivalent on parallelogram meshes. Computational results indicate that both techniques perform extremely well on smooth quadrilateral meshes. On nonsmooth meshes, the preserving technique retains its excellent performance while the CP technique degrades. The reasons for this degradation are discussed. Although the SP scheme has proven to be generally effective on quadrilateral meshes in x-y geometry, it is not expected to be effective for quadrilaterals in r-z geometry or for hexahedra in three-dimensional Cartesian geometry. Thus, a full lumping procedure for general nonorthogonal meshes that possesses all of the desired properties has yet to be found. For reasons that are discussed, it appears unlikely that such a procedure exists.

Interface reconstruction with least-squares fit and split advection in three-dimensional Cartesian geometry

In this paper we present and analyze different volume-of-fluid (VOF) reconstruction and advection algorithms that approximate the interface separating two immiscible fluids in the three-dimensional space. The paper describes the improvement of the reconstruction when a least-square fit algorithm, which minimizes a distance functional, is applied. Its performance is tested for several smooth surfaces against other simple reconstruction methods. Then Eulerian, Lagrangian and mixed split advection schemes are presented and analyzed. In particular, one advection method is discussed that conserves mass exactly for a divergence-free velocity field, thus allowing computations to machine precision.

An Efficient Exponential Directional Iterative Differencing Scheme for Three-Dimensional Sn Computations in XYZ Geometry

A new exponential spatial differencing scheme based on zeroth spatial transport moments, the exponential directional iterative (EDI) Sn scheme for three-dimensional (3-D) Cartesian geometry, is presented. The EDI scheme is a logical extension of the positive, efficient exponential directional weighted (EDW) method used in the PENTRAN parallel Sn solver in an adaptive differencing strategy. The EDI scheme uses EDW-rendered exponential coefficients as initial values to begin a fixed-point iteration to refine exponential coefficients. Iterative refinement of these coefficients typically converged in fewer than four fixed-point iterations per ordinate, and yielded more accurate angular fluxes compared to other schemes tested. Overall, the EDI scheme is an order of magnitude more accurate than EDW, and two orders of magnitude more accurate than the legacy diamond zero (DZ) scheme for a given mesh. EDI is therefore a good candidate for a fourth-level scheme in the PENTRAN adaptive sequence. The 3-D Cartesian computational cost of EDI was ˜20% more than EDW, and only ˜40% more than DZ. Thus, EDI renders increased accuracy using zeroth spatial transport moments in a straightforward manner for any 3-D Cartesian code. More evaluation is ongoing to determine suitability in an upgraded adaptive differencing sequence algorithm in PENTRAN.

Interface Conditions for Spherical Harmonics Methods

A set of interface conditions is derived rigorously for the general spherical harmonics solution of the Boltzmann transport equation in three-dimensional Cartesian geometry. The derivation builds upon earlier work of Davidson and Rumyantsev to arrive at sets of interface conditions applicable to both even- and odd-order N spherical harmonics approximations. The exact set of conditions is compared to the approximate set currently employed in the odd-order N variational nodal code VARIANT, and the differences in accuracy and computational effort are summarized. The exact interface conditions are necessary for first-order implementations of spherical harmonics methods.

A Compatible, Energy and Symmetry Preserving Lagrangian Hydrodynamics Algorithm in Three-Dimensional Cartesian Geometry

This work presents a numerical algorithm for the solution of fluid dynamics problems with moderate to high speed flow in three dimensions. Cartesian geometry is chosen owing to the fact that in this coordinate system no curvature terms are present that break the conservation law structure of the fluid equations. Written in Lagrangian form, these equations are discretized utilizing compatible, control volume differencing with a staggered-grid placement of the spatial variables. The concept of “compatibility” means that the forces used in the momentum equation to advance velocity are also incorporated into the internal energy equation so that these equations together define the total energy as a quantity that is exactly conserved in time in discrete form. Multiple pressures are utilized in each zone; they produce forces that resist spurious vorticity generation. This difficulty can severely limit the utility of the Lagrangian formulation in two dimensions and make this representation otherwise virtually useless in three dimensions. An edge-centered artificial viscosity whose magnitude is regulated by local velocity gradients is used to capture shocks. The particular difficulty of exactly preserving one-dimensional spherical symmetry in three-dimensional geometry is solved. This problem has both practical and pedagogical significance. The algorithm is suitable for both structured and unstructured grids. Limitations that symmetry preservation imposes on the latter type of grids are delineated.