One-dimensional Spherical Geometry Research Articles

4th-Order-SENS: A Software Module for Efficient and Exact 4th-Order Sensitivity Analysis of Neutron Transport

This work presents a software module called 4th-Order-SENS, which enables the efficient computation of exactly obtained expressions for all sensitivities, up to and including the 4th order, of a functional of the particle flux (e.g., the leakage of particles out of a body) with respect to nuclear parameters (total, scattering, and fission cross sections; nu, chi, sources; and number densities) for systems modeled by the neutron transport equation. The 4th-Order-SENS module implements the nth-Order Comprehensive Adjoint Sensitivity Analysis Methodology for Linear Systems (nth-CASAM-L), which is the only practically implementable methodology for obtaining the exact expressions of arbitrarily high-order sensitivities of model responses to model parameters, for response-coupled forward/adjoint large-scale linear systems. In addition to presenting the equations that are solved to obtain the 1st-order through 4th-order sensitivities, this work also describes the components of the module 4th-Order-SENS, including the user interface, input file, output files, and several independent code verification capabilities using symmetries and/or finite-difference formulas. The 4th-Order-SENS module is written in Python and Fortran and runs on Linux platforms. Several illustrative applications involving fixed-source problems in one-dimensional spherical and slab geometries are also presented.

Permeation grouting in soils: numerical discussion of a simplified analytical approach

Permeation grouting – injections at low-pressure values of either microfine cements or chemical products – is frequently adopted to increase the mechanical/hydraulic properties of soils, and standard design approaches are currently either empirical or based on simplified analytical solutions. In this paper, some fundamental hypotheses of these analytical solutions are discussed by carrying out a campaign of finite-element numerical analyses, in which the injection phase in a water-saturated soil is analysed, a Newtonian rheology for the grout is implemented and the hypothesis of immiscibility for the two liquids is assumed. The effect of injection source geometry is discussed, as well as the role of gravity and capillarity. The authors analyse the conditions, in terms of injection flow rate, grout viscosity, soil intrinsic permeability and retention curve, under which the analytical solutions provide reliable results. The numerical results have been compared with the simplified analytical solution herein derived for a one-dimensional spherical geometry, in terms of the ‘characteristic curves’ for the system: the relationship between injection pressure/grout front advancement and injection time.

Dark Off-limb Gap: Manifestation of a Temperature Minimum and the Dynamic Nature of the Chromosphere

We study off-limb emission of the lower solar atmosphere using high-resolution imaging spectroscopy in the Hβ and Ca ii 8542 Å lines obtained with the CHROMospheric Imaging Spectrometer (CHROMIS) and the CRisp Imaging SpectroPolarimeter (CRISP) on the Swedish 1 m Solar Telescope. The Hβ line-wing images show the dark intensity gap between the photospheric limb and chromosphere, which is absent in the Ca ii images. We calculate synthetic spectra of the off-limb emissions with the RH code in one-dimensional spherical geometry and find good agreement with the observations. The analysis of synthetic line profiles shows that the gap in the Hβ line-wing images maps the temperature minimum region between the photosphere and chromosphere due to the well-known opacity and emissivity gap of Balmer lines in this layer. However, the observed gap is detected farther from the line core in the outer line-wing positions than in the synthetic profiles. We found that an increased microturbulence in the model chromosphere is needed to reproduce the dark gap in the outer line wing, suggesting that the observed Hβ gap is the manifestation of the temperature minimum and the dynamic nature of the solar chromosphere. The temperature minimum produces a small enhancement in synthetic Ca ii line-wing intensities. Observed off-limb Ca ii line-wing emissions show similar enhancement below the temperature minimum layer near the edge of the photospheric limb.

Investigation on the fusion reaction rate of deuterium and tritium under heterogeneous mixing

Mixing between shell material and gas fuel, caused by hydrodynamic instability, isolated defects, or kinetic effects, is the key to understand the degradation of implosion performance in the research of inertial confinement fusion. Understanding the mixing mechanism and reducing its impact is of extreme importance to achieve the ignition and high gain. The impact of mixing morphology on thermonuclear reaction rate in sub grid level has gradually attracted people’s attention in recent years due to its direct influence on burn rate and fusion process, the study on physical model of thermonuclear reaction rate in different mix morphology has important scientific significance and application value. In the paper, the dependence of thermonuclear reaction rate on mass distribution of different fuel concentrations at sub grid scale is derived. Based on thermodynamic equilibrium and ideal gas equation of state, the physical law of the evolution of the thermonuclear reaction rate with mix morphology under the dominance of diffusion mixing is revealed through analytical formula and numerical solution of diffusion equation in one-dimensional spherical geometry. It is convinced that the mixing amount directly affects the thermonuclear reaction rate by mainly affecting the volume fraction of the fuel, and the mixing diffusion time determined by heterogeneous mixing scale and diffusion coefficient directly affects the evolution behavior of the thermonuclear reaction rate. Furthermore, based on mutual diffusion coefficient obtained from direct simulation of diffusion process by Monte Carlo method, the difference of impact to thermonuclear reaction rate for low-<i>Z</i> Carbon and high-<i>Z</i> gold mixing is quantitatively investigated. Heterogeneous mix size with 0.1 μm, 0.01 μm respectively for the low-<i>Z</i> and high-<i>Z</i> mixing can be treated as atomic mix in burn rate aspect, and heterogeneous mix size with 10 μm, 1 μm respectively for the low-<i>Z</i> and high-<i>Z</i> mixing can be treated as ideal chunk mix in burn rate aspect, and heterogeneous mix size in the middle state needs to be evaluated by using the heterogeneous mixing model of thermonuclear reaction rate in the paper. Finally, the physical model is compared with 3D simulation results of the heterogeneous mixing effect experiment called “MARBLE Campaign” carried out on OMEGA laser facility, which is designed as a separated reactant experiments and capsules are filled with deuterated foam and HT gas pores of different size, covering typical mix morphology from atomic mix to chunk mix, which validate the reliability of the theoretical evaluation about the evolution of mixing morphology and its impact to thermonuclear reaction rate. This work is significant for the design and improvement of inertial confinement fusion mixing effect experiment in China.

Inverse Method for Controlling Pure Material Solidification in Spherical Geometry

In this study, we present the control of the solidification process of a phase-changing, pure material described in one-dimensional spherical geometry. We used an inverse global descent method in which the gradient and the adjoint equation are constructed in continuous variables of time and space. The control variable is the temperature at the fixed boundary of the solid domain. For the desired solidification front, the control was determined using information on the heat flux deduced by heat balance. The numerical resolution was based on a finite difference method in a physical domain with a moving grid related to the evolving solidification front with time. The developed numerical model was validated using an exact built solution. The numerical results of the control problem are presented for both the exact and noisy data cases. For the noisy data, a regularization method was applied. In the case of the exactdata, a rapid control determination was achieved except for time steps near the end. The random errors effects in bruited data were considerably reduced by regularization.

Plasma viscosity in spherical ICF implosion simulations

Inertial confinement fusion (ICF) hydrodynamic codes often ignore the effects of viscosity though recent research indicates plasma viscosity and mixing by classical transport processes may have a substantial impact on implosion dynamics. A Lagrangian hydrodynamic code in one-dimensional spherical geometry with plasma viscosity and mass transport, and including a three temperature model for ions, electrons, and radiation treated in a gray radiation diffusion approximation, is used to study differences between ICF implosions with and without plasma viscosity and to examine the role of artificial viscosity in a Lagrangian implosion simulation. It was found that plasma viscosity has substantial impacts on ICF shock dynamics characterized by shock burn timing, maximum burn temperatures, fuel compression, and time history of neutron production rates. Plasma viscosity reduces the need for artificial viscosity to maintain numerical stability in the Lagrangian formulation and this study suggests that artificial viscosity may provide an unphysical stability in implosion simulations.

Plasma viscosity with mass transport in spherical inertial confinement fusion implosion simulations

The effects of viscosity and small-scale atomic-level mixing on plasmas in inertial confinement fusion (ICF) currently represent challenges in ICF research. Many current ICF hydrodynamic codes ignore the effects of viscosity though recent research indicates viscosity and mixing by classical transport processes may have a substantial impact on implosion dynamics. We have implemented a Lagrangian hydrodynamic code in one-dimensional spherical geometry with plasma viscosity and mass transport and including a three temperature model for ions, electrons, and radiation treated in a gray radiation diffusion approximation. The code is used to study ICF implosion differences with and without plasma viscosity and to determine the impacts of viscosity on temperature histories and neutron yield. It was found that plasma viscosity has substantial impacts on ICF shock dynamics characterized by shock burn timing, maximum burn temperatures, convergence ratio, and time history of neutron production rates. Plasma viscosity reduces the need for artificial viscosity to maintain numerical stability in the Lagrangian formulation and also modifies the flux-limiting needed for electron thermal conduction.

Neutron transport in Eulerian coordinates with bulk material motion

A consistent, numerically stable algorithm for the solution of the neutron transport equation in the presence of a moving material background is presented for one-dimensional spherical geometry. Manufactured solutions are used to demonstrate the correctness and stability of our numerical algorithm. The importance of including moving material corrections is shown for the r-process in proto-neutron stars.

T 1 and U 1 approximations to neutron transport equation in one-dimensional spherical geometry

Abstract The diffusion lengths for one-speed neutrons are calculated by solving transport equation in one-dimensional spherical geometry. The pseudo-slab transport equation is obtained using a reasonable approach. The neutron angular flux is expanded in a series of both the Chebyshev polynomials of first and second kinds. Then analytic and numerical results for asymptotic relaxation length are obtained by applying first order approximations to the transport equation and they are compared with each other.

One-dimensional spherical ray-tracing extended to account for flat and cylindrical shields

It is often desirable to compute the uncollided component (direct from the source) of a detected particle flux using some form of ray-tracing, which in one-dimensional spherical geometries is extremely economical. A simple method for treating a flat shield exactly, within the context of spherical ray-tracing, is presented. A method for treating a cylindrical shield approximately, using the average value of the azimuthal angle for each value of the polar angle that is used in the spherical ray-tracing, is presented. The source remains spherical. The methods are verified numerically with a test problem that also shows the importance of these corrections.

Nonspherical Perturbations of Spherical Geometries in Transport Theory

It is often desirable to solve radiation transport problems in one-dimensional spherical geometries even if the actual object being modeled is not spherical. It may be possible to use perturbation theory to account for the difference between the real multidimensional system and the spherical approximation. This idea is tested using uncollided as well as multigroup inhomogeneous transport problems with upscattering. Asymmetric and nonuniform perturbations are made to the shielding (not the source) of spherical geometries, including transformations from a sphere to a cube (the surface transformation function is derived), and Schwinger, Roussopolos, and combined perturbation estimates are applied. For uncollided fluxes, perturbation theory, particularly the Schwinger estimate, worked very well when the response of interest was the flux measured at a symmetric spherical 4π detector external to the geometry, but perturbation theory did not work well when the response of interest was the flux measured at a single external point (unless extra care was taken to account for geometric effects). For neutron-induced gamma-ray line fluxes, the Roussopolos estimate worked well when the response of interest was the flux measured at an external 4π detector or an external point detector.

Chebyshev polynomials expansion method for solving the one-dimensional transport equation in spherical geometry

Abstract Using certain well-known properties of Chebyshev polynomials, an efficient analytical approach to evaluate the Boltzmann equation is presented in one-dimensional spherical geometry. The procedure is based on the expansion of the angular flux in terms of a series of Chebyshev polynomials.

Moments-Based Versions of the Spherical Geometry Method of Tubes in Thick, Diffusive Regions

Characteristic methods are widely known to be very accurate approaches to the solution of numerical transport problems. These methods are most often used for neutron transport applications (i.e., lattice physics calculations) where spatial cells are of intermediate optical thickness [O(1) to O(100) mean free paths, depending on the energy group] and materials are not exceptionally highly scattering (scattering ratios < 0.999). There has been interest in using characteristic methods for radiative transfer applications, which often involve very optically thick and diffusive regions. Previous work has involved analyses of families of Cartesian geometry characteristic methods in optically thick and diffusive regions. There is a significant body of work in the Russian literature on curvilinear geometry characteristic methods, but very few analyses of their behavior in thick diffusive regions have been published. In this paper we develop two new members of a family of one-dimensional spherical geometry characteristic methods—the method of tubes. These new methods are similar to traditional slab geometry characteristics methods in that they utilize spatial moments of the transport equation in each cell to generate the data used in the representation of the total source (scattering source plus external source). We present the results of an asymptotic analysis of these methods to predict their behavior in the thick diffusion limit, and we compare these predictions with numerical results from several test problems. This analysis shows that the constant source (step) method behaves very poorly in the diffusion limit, but that the linear source method is accurate in this physical regime.

Simulations of Cyclic Voltammetric and Chronoamperometric Electrode Responses at a Disk Electrode Using Combinations of Spherical and Cylindrical Electrode Geometries

Using geometric models based on one-dimensional transport at spheres and cylinders, three methods for improving the simulation of voltammetric behavior of a disk electrode have been explored. One method is based on the common assumption of equivalency between the limiting currents for a disk and a hemisphere under steady-state diffusion conditions. The second method involves the use of a partial-sphere geometry which is a better approximation that is suitable at the extreme diffusional limits achievable at a disk electrode of fully planar and steady-state transport. The third method, which is generally applicable, is a further refinement that uses a combination of appropriate one-dimensional spherical and cylindrical geometries. The results demonstrate that reasonably accurate approximations of disk behavior for several reaction mechanisms can be achieved in a fraction of the time required to compute the more rigorous two-dimensional model. We propose that the approximation serve primarily as a fast way to explore system behavior and establish approximate values of the relevant parameters. More accurate computations can then be performed using the two-dimensional model.

Feedback from multiple supernova explosions inside a wind-blown bubble

We study the evolution of multiple supernova (SN) explosions inside a pre-exiting cavity blown by winds from massive progenitor stars. Hydrodynamic simulations in one-dimensional spherical geometry, including radiative cooling and thermal conduction, are carried out to follow first the development of the wind-blown bubble during the main sequence and then the evolution of the SN-driven bubble. We find the size and mass of the SN-driven bubble shell depend on the structure of the pre-existing wind bubble as well as the SN explosion energy E SN (= N SN10 51 ergs). The hot cavity inside the bubble is 2–3 times bigger in volume and hotter than that of a bubble created by SNe exploded in a uniform interstellar medium (ISM). For an association with 10 massive stars in the average ISM, the SN-driven shell has an outer radius of R ss ≈ ( 85 pc ) N SN 0.1 and a mass of M ss ≈ ( 10 4.8 M ⊙ ) N SN 0.3 at 10 6 years after the explosion. By that time most of the explosion energy is lost via radiative cooling, while ≲10% remains as kinetic energy and ∼10% as thermal energy. We also calculate the total integrated spectrum of diffuse radiation emitted by the shock-heated gas of the SN bubble. Total number of H Lymann-limit photons scales roughly as Φ 13.6 ≈ 10 61 N SN and those photons carry away 20–55% of the explosion energy. For the models with 0.1 solar metalicity, the radiative energy loss is smaller and the fraction of non-ionizing photons is larger, compared to those with solar metalicity. We conclude the photoionization/heating by diffuse radiation is the most dominant form of feedback from SN explosions into the surrounding medium.

The Search for Superconvergence in Spherical Harmonics Approximations

The occurrence of superconvergence in various first-order spherical harmonics approximations of the neutral particle transport equation is being investigated. Superconvergence refers to the added accuracy gained in evaluating the solution of the transport equation at optimally chosen base points of the finite element trial functions. It has been observed that this phenomenon is happening when primal and dual discretizations in space and angle lead to the same numerical result, a property also referred as primal-dual agreement. A systematic search is presented for primal-dual agreement on one-dimensional slab, tube, and spherical geometries and on Cartesian two-dimensional geometries based on complete and simplified Pn approximations. Primal-dual agreement was successfully obtained in all Cartesian geometries but not in tube and spherical geometries, due to the angular redistribution term.

Numerical studies of diffusive shock acceleration at spherical shocks

We have developed a cosmic ray (CR) shock code in one-dimensional spherical geometry with which the particle distribution, the gas flow and their nonlinear interaction can be followed numerically in a frame comoving with an expanding shock. In order to accommodate a very wide dynamic range of diffusion length scales in the CR shock problem, we have incorporated subzone shock tracking and adaptive mesh refinement techniques. We find the spatial grid resolution required for numerical convergence is less stringent in this code compared to typical, fixed-grid Eulerian codes. The improved convergence behavior derives from maintaining the shock discontinuity inside the same grid zone in the comoving code. That feature improves numerical estimates of the compression rate experienced by CRs crossing the subshock compared to codes that allow the subshock to drift on the grid. Using this code with a Bohm-like diffusion model we have calculated the CR acceleration and the nonlinear feedback at supernova remnant shocks during the Sedov–Taylor stage. Similarly to plane-parallel shocks, with an adopted thermal leakage injection model, about 10 −3 of the particles that pass through the shock and up to 60% of the explosion energy are transferred to the CR component. These results are in good agreement with previous nonlinear spherical CR shock calculations of Berezhko and collaborators.

A Numerical Simulation on the Solar-Terrestrial Transit Time of Successive CMEs during November 4-5, 1998

The solar-terrestrial transit process of three successive coronal mass ejections (CMEs) during November 4-5, 1998 has been investigated numerically in one-dimensional spherical geometry. These CMEs interact with each other while they are propagating in interplanetary space and finally form a “Complex Ejecta”. A Harten's total variation diminishing (TVD) scheme is applied to solve magnetohydrodynamic (MHD) equations numerically, starting from an ambient solar wind equilibrium, with appropriate dimensionless gravity parameter α, plasma beta β, and gas polytropic index γ The equilibrium is consistent in solar wind speed υr, proton number density Np, and the ratio of proton thermal pressure to magnetic pressure βp with the observation of ACE spacecraft at Lagrange point (L1). Merely velocity pulse is introduced in the numerical computation, whose amplitude and duration are determined by observation data of Lasco/C2, GOES, LEAR combined with CME's “Cone Model” proposed by Michalek et al. The results show that the differences of two shock arrival times (SATs) between the numerical calculation and ACE observation are 3 and 4 hours respectively. Therefore the numerical model proposed in this paper can estimate SAT and rough shock intensity formed by successive CMEs evolving in interplanetary space and suggests a potential application in SAT prediction for space weather.

Singular Solutions, Integral Transport Theory, and the Sn Method

The integral transport equation clearly indicates that the angular flux in a void is constant along each characteristic. Yet, simple arguments can be used to demonstrate that there exist angular flux solutions in voids that have a delta-function angular dependence and a nonconstant spatial dependence. Such solutions can appear to be nonconstant along a characteristic. Using a simple example problem, we demonstrate that such solutions represent the limit of a continuous sequence of nonsingular solutions, each of which is constant along every characteristic. We also show that care must be taken in applying the integral transport equation to singular problems of this type because erroneous solutions are easily obtained. Two reliable approaches for obtaining proper solutions are presented. We also show that the differential form of the transport equation in one-dimensional spherical geometry requires less care than the integral form of the transport equation for problems of this type. Finally, we discuss the applicability of the Sn method to problems in curvilinear geometries with singular solutions of this type.

A Comparison of Angular Difference Schemes for One-Dimensional Spherical Geometry SN Equations

To investigate errors caused by angular differencing in approximating the streaming terms of the transport equation, five different approximations are evaluated for three test problems in one-dimensional spherical geometry. The following schemes are compared: diamond, special truncation error minimizing weighted diamond, linear continuous (the original SN scheme), linear discontinuous, and new quadratic continuous. To isolate errors caused by angular differencing, the approximations are derived from the transport equation without spatial differencing, and the resulting coupled ordinary differential equations (ODEs) are solved with an ODE solver. Results from the approximations are compared with analytic solutions derived for two-region purely absorbing spheres. Most of the approximations are derived by taking moments of the conservation form of the transport equation. The quadratic continuous approximation is derived taking the zeroth moment of both the transport equation and the first angular derivative of the transport equation. The advantages of this approach are described. In all of the approximations, the desirability is shown of using an initializing computation of the μ = -1 angular flux to correctly compute the central flux and of having a difference approximation that ensures this central flux is the same for all directions. The behavior of the standard discrete ordinates equations in the diffusion limit is reviewed, and the linear and quadratic continuous approximations are shown to have the correct diffusion limit if an equal interval discrete quadrature is used.In all three test problems, the weighted diamond difference approximation has smaller maximum and average relative flux errors than the diamond or the linear continuous difference approximations. The quadratic continuous approximation and the linear discontinuous approximation are both more accurate than the other approximations, and the quadratic continuous approximation has a decided edge over the linear discontinuous approximation in relative flux errors. The diamond, weighted diamond, and linear continuous approximations show quadratic system absorption and system leakage error reduction behavior with increasing N. The linear discontinuous and quadratic continuous approximations show fourth-order error reduction in these quantities. In one of the two-region test problems, the slope of the exact angular flux changes from nearly vertical to nearly horizontal at those points in the exterior region at which the interior region source just becomes visible. At these spatial points, errors in the continuous approximations propagate to each successive outgoing direction, leading to an oscillatory spatial error. The discontinuous approximation does not propagate these errors, although errors near the point of rapid slope change are larger than in the quadratic continuous approximation.