Manoj K. Prasad

3D unstructured mesh ALE hydrodynamics with the upwind discontinuous finite element method

Abstract We describe a numerical scheme to solve 3D Arbitrary Lagrangian-Eulerian (ALE) hydrodynamics on an unstructured mesh using discontinuous finite element space and an explicit Runge-Kutta time discretization. This scheme combines the accuracy of a higher-order Godunov scheme with the unstructured mesh capabilities of finite elements that can be explicitly evolved in time. The spatial discretization uses trilinear isoparametric elements (tetrahedrons, pyramids, prisms and hexahedrons) in which the primitive variables (mass density, velocity and pressure) are piecewise trilinear. Upwinding is achieved by using Roes characteristic decomposition of the inter-element boundary flux depending on the sign of characteristic wave speeds. The characteristics are evaluated at the Roe average, of variables on both sides of the inter-element boundary, for a general equation of state. An explicit second order Runge-Kutta time stepping is used for the time discretization. To capture shocks, we have generalized van Leers 1D nonlinear minmod slope limiter to 3D using a quadratic programming scheme. For very strong shocks we find it necessary to supplement this with a Godunov stabilization where the trilinear representation of the variables is reduced to its constant average value. The resulting numerical scheme has been tested on a variety of problems relevant to ICF (inertial confinement fusion) target design and appears to be robust. It accurately captures shocks and contact discontinuities without unstable oscillations and has second-order accuracy in smooth regions. Object-oriented programming with the C++ programming language was used to implement our numerical scheme. The object-oriented design allows us to remove the complexities of an unstructured mesh from the basic physics modules and thereby enables efficient code development.

A simple and fast method for computing the relativistic Compton scattering kernel for radiative transfer

Abstract The Klein-Nishina differential cross section averaged over a relativistic Maxwellian electron distribution is analytically reduced to a single integral, which can then be rapidly evaluated in a variety of ways. A particularly fast method for numerically computing this single integral is presented. This is, to our knowledge, the first correct computation of the Compton scattering kernel.

Statistical Theory of Fission Chains and Generalized Poisson Neutron Counting Distributions

Abstract The neutron counting probability distribution for a multiplying medium was shown by Hage and Cifarelli to be a generalized Poisson distribution that depends on the fission chain number distribution. An analytic formula is obtained for this number distribution, the probability to produce a number of neutrons in a fission chain. The formula depends on the probability that a fission spectrum neutron induces a subsequent fission and depends on the probability distribution for a specific number of neutrons to be produced in an individual induced fission. The formula is an exact solution to a functional equation due to Böhnel for the probability generating function. The Böhnel equation is derived as the t →∞ limit of a rate equation for a neutron population generating function, related to a rate equation studied by Feynman. The Böhnel equation is also shown to be a fixed point of an iteration problem, related to one studied by Hawkins and Ulam, where the iteration generates the chain a generation at a time. The discrete iteration problem is shown to be connected to the continuous time evolution of the chain. An explicit solution for the time evolution of the chain is given in the simplified approximation where at most two neutrons are created by an induced fission. The t →∞ limit of this equation gives a simple analytic expression for the solution to the Böhnel equation in this approximation. A generalized Poisson counting distribution constructed from the theoretical fission chain probability number distribution is compared to experimental data for a multiplying Pu sample.

The radiation-hydrodynamic ICF3D code

Abstract We describe the 3D high temperature plasma simulation computer code ICF3D which is being developed at the Lawrence Livermore National Laboratory. The code is portable; it runs on a variety of platforms: uniprocessors, SMPs, and MPPs. It parallelizes by decomposing physical space into disjoint subdomains and relies on message passing libraries such as MPI. ICF3D is written in the object oriented programming language C++. The mesh is unstructured and consists of a collection of hexahedra, prisms, pyramids, and/or tetrahedra. The hydrodynamics is modeled by the discontinuous finite element method which allows a natural representation of inherently discontinuous phenomena such as shocks. Continuous processes such as diffusion are modeled by conventional finite element methods. ICF3D is modular and consists of separate equation-of-state, hydrodynamic, heat conduction, and multi-group radiation transport (diffusion approximation) packages. We present results on problems relevant to Inertial Confinement Fusion which are obtained on a variety of computers, uniprocessors and MPPs.

Evaluation of integrals of the Compton scattering cross-section

Abstract Exact formulae are derived for various integrals of the Compton scattering cross-section. The interaction kernel is integrated over outgoing photon frequency and direction, and over a relativistic Maxwellian distribution for the electrons. The total Compton cross-section, the energy exchange rate, and the transport mean free path are thereby expressed in terms of single integrals of analytic functions. In addition, these integrals produce simple analytic expressions in the limiting cases of either small or large frequency or electron temperature. A numerical method based on Gaussian quadrature is used to compute the transport mean free path. A comparison with previously published results is presented.

Time Interval Distributions and the Rossi Correlation Function

Abstract For material spontaneously generating fission chains, the arrival times of neutron and gamma-ray counts create a clustering pattern distinctly different from a random source. A theory for the time interval distribution between counts is given. As well as the distribution of nearest-neighbor counts, we give the general distributions for all n’th-neighbor intervals. The sum of these distributions gives the Rossi correlation function. This theory supplies the direct link between the experimentally measured quantities and the theory of the Rossi correlation function.

Diffusion coefficient for the Compton Fokker-Planck equation☆

Abstract An exact analytical formula for the diffusion coefficient of the Compton Fokker-Planck equation is derived. The formula is valid for arbitrary values of the electron temperature and photon energy. For applications in production-level radiation transport codes, a fast numerical method to compute the coefficient is presented.

Time Evolving Fission Chain Theory and Fast Neutron and Gamma-Ray Counting Distributions

Abstract We solve a simple theoretical model of time evolving fission chains due to Feynman that generalizes and asymptotically approaches the point model theory. The point model theory has been used to analyze thermal neutron counting data. This extension of the theory underlies fast counting data for both neutrons and gamma rays from metal systems. Fast neutron and gamma-ray counting is now possible using liquid scintillator arrays with nanosecond time resolution. For individual fission chains, the differential equations describing three correlated probability distributions are solved: the time-dependent internal neutron population, accumulation of fissions in time, and accumulation of leaked neutrons in time. Explicit analytic formulas are given for correlated moments of the time evolving chain populations. The equations for random time gate fast neutron and gamma-ray counting distributions, due to randomly initiated chains, are presented. Correlated moment equations are given for both random time gate and triggered time gate counting. Explicit formulas for all correlated moments are given up to triple order, for all combinations of correlated fast neutrons and gamma rays. The nonlinear differential equations for probabilities for time dependent fission chain populations have a remarkably simple Monte Carlo realization. A Monte Carlo code was developed for this theory and is shown to statistically realize the solutions to the fission chain theory probability distributions. Combined with random initiation of chains and detection of external quanta, the Monte Carlo code generates time tagged data for neutron and gamma-ray counting and from these data the counting distributions.

Neutron Time Interval Distributions with Background Neutrons

Abstract A single cosmic ray air shower event can produce multiple neutrons. The arrival times of neutron counts from such an event creates a clustering pattern distinctly different from random sources. A theory for the time interval distribution between neutron counts from both a correlated source and cosmic ray air showers is given and a method is developed to compute the probability distributions for a cosmic ray air shower to create detected neutrons.

3D unstructured mesh ALE hydrodynamics with the upwind discontinuous galerkin method

The authors describe a numerical scheme to solve 3D Arbitrary Lagrangian-Eulerian (ALE) hydrodynamics on an unstructured mesh using a discontinuous Galerkin method (DGM) and an explicit Runge-Kutta time discretization. Upwinding is achieved through Roes linearized Riemann solver with the Harten-Hyman entropy fix. For stabilization, a 3D quadratic programming generalization of van Leers 1D minmod slope limiter is used along with a Lapidus type artificial viscosity. This DGM scheme has been tested on a variety of hydrodynamic test problems and appears to be robust making it the basis for the integrated 3D inertial confinement fusion modeling code (ICF3D). For efficient code development, they use C++ object oriented programming to easily separate the complexities of an unstructured mesh from the basic physics modules. ICF3D is fully parallelized using domain decomposition and the MPI message passing library. It is fully portable. It runs on uniprocessor workstations and massively parallel platforms with distributed and shared memory.