Nonlinear acceleration methods for even parity neutron transport, william martin. An improved variational nodal method for the solution of. When a nucleus contains an odd number of both particle types, it is nearly always unstable. Nuclear engineering etds engineering etds university. With changes only in the interaction laws contained in the cross. A modular nodal method is developed for solving the neutron transport equation by using the spherical harmonics approximation in two dimensional cartesian coordinates. Discrete ordinates in 2d and 3d geometries with fe mixed mesh capabilities these second order methods have been implemented on. Proteussn is a threedimensional deterministic neutron transport code which solves the second order formulation of the neutron transport equation. Multilevel acceleration of neutron transport calculations approved by. Coarse mesh methods for the transport calculation in. The first one describes local behavior of the density at the cell level. The interface separating the models is chosen so that the diffusive regime holds in its vicinity to avoid the calculation of boundary or interface layers.
The second order even parity form of the transport equation is discretized using the continuous galerkin finite element method in. The principal application is to the deeppenetration transport of neutrons andor photons. Numerical solutions to neutron transport equation are required in reactor core. Abstracta variational finite elementspherical harmonics method is presented for the solution of the even parity multigroup equations with anisotropic scattering and sources. Evenparity, or second order, neutron transport has been used in a limited capacity historically due to advantages and popularity of other deterministic methods. The even parity equation is solved on the fine mesh using moc, while the odd parity. Siam journal on mathematical analysis siam society for. The second, developed here uses a variational principle as a point of departure for the application of finite elements to neutron transport problems. A modular nodal method for solving the neutron transport. Spherical harmonic method in 1d, 2d and 3d geometries with fe mixed mesh capabilities sn2nd. Transport approximations the spectral element method is well suited for the second order form of the transport equation. The spherical harmonics approximation is based upon the second order even parity form of the neutron transport equation.
In the remainder of this course we will assume that in any reaction, we know the probability of interaction of a neutron with a nucleus for any given neutron. Fpga hardware acceleration for high performance neutron transport. Numerical methods in the theory of neutron transport book. Coupling of transport and diffusion models in linear. The first part of the book covers basic reactor physics, including, but not limited to nuclear reaction data, neutron diffusion theory, reactor. The methods for the numerical solution of the neutron transport equation can be divided. It is a solution of a heterogeneous transport problem with. While it is possible to replace the even parity methodology in proteussn with a supg or gls scheme, this does not. The treatment of the neutron transport equation by the evenodd parity formalism. Finite element investigations for the construction of. Computational methods of neutron transport book osti.
The results for the linear and nonlinear case serve as the basis for further research into the application in a full threedimensional sphericalharmonics evenparity transport code. This spin effect finds expression in the fact that nuclei with an even number of protons and an even number of neutrons are very stable thanks to the occurrence of paired spin. Evenparity neutron transport equation eigenvalue search nonlinear systems 1 introduction the keigenvalue calculation in criticality problems has traditionallyutilizedtheclassicalpoweriterationmethod which has slow convergence order. So, let us consider the withingroup transport equation for the neutron. Development of code, pnfent, based on using finite. Convergence acceleration methods for even parity transport were developed that have the potential to speed up transport calculations and provide a natural avenue for an implicitly coupled multiphysics code. A finite analytic not finitedifference scheme is developed in the characteristic value for the solution of even and odd parity kinetic equations of neutron and photon transport with algebraic and centered forms of the scattering integral in onedimensional problems with the symmetry of a plane layer, cylinder, and sphere. Read even and odd parity kinetic equations of particle transport. If the even parity form of the transport equation is used, the spherical harmonics approach provides a set of second order differential equations. The particle scattering integral of even and odd parity transport equations is converted into a nonlinear algebraic form and into a centered form. Nodal diffusion and transport methods are formulated variationally in terms of the evenparity form of the neutron transport equation and. It is a nonlinear integrodifferential equation for the phase space density of the molecules of a dilute gas. The boltzmann transport equation, governing the neutron distribution in a nuclear reactor leads, by using the vladimirov method given in 12, to the even parity second order transport equation.
The code is able to solve multidimensional forward and adjoint neutron transport equation over an arbitrary geometry using linear or high order elements. A goaloriented and selfadaptive mesh refinement approach. Starting from this result we have first more thoroughly investigated the order of approximation, with respect to the time variable, arising. Coupled neutronics thermal hydraulics event thermix. Exppg yloiting concurrency at the petascale in neutron. Even and oddparity kinetic equations of particle transport. The third, revised edition of this popular textbook and reference, which has been translated into russian and chinese, expands the comprehensive and balanced coverage of nuclear reactor physics to include recent advances in understanding of this topic.
Once moved into the nonlinear solution scheme, the implicit coupling of the convergence accelerated transport method into codes for. International conference on the physics of reactors 2010. New insights into numerical solutions of the even transport equation in twodimensional xy geometry noh, taewan. Although it has not been widely used, the numerical advantages for evenparity transport are plenty, especially when. Neutron transport equations, newtonkrylovschwarz, mesh partitioning, workload balancing, parallel computation, multilevel domain decomposition methods.
Although this angular dependency can be approximated by various approaches, the spherical harmonics approach offers some advantages. The withingroup even parity neutron transport equation is formulated with complex angular and spatial trial functions and with a complex buckling approximation. We consider the homogenization of the criticality eigenvalue problem for the even parity flux of neutron transport in a domain with isotropic and periodically oscillating coefficients. Algebraic and centered forms of the scattering integral, mathematical models and computer simulations on deepdyve, the largest online rental service for scholarly research with thousands of academic publications available at your fingertips. A conjoint variational formulation based on discontinuous finite elements approach for pn neutron transport equation has been presented. This third, completely revised edition of the textbook retains the proven concept of complete and balanced coverage of the topic. Finite element investigations have been carried out to construct composite solutions of transport problems. A novel neutron recoil spectrometer concept utilizing heavyion recoils and time and spatiallyresolved sensor, joel long. Even parity neutron transport event a variational finite elementspherical harmonics method for solving even parity multigroup neutron transport equations with anisotropic scattering possibility to deal with complex geometries 2d3d even parity equation reduces.
Nonlinear acceleration methods for evenparity neutron. Park acceleration technique using krylov subspace methods fo r 2d arbitrary geometry characteristics. The particle scattering integral of even and oddparity transport equations is converted into a nonlinear algebraic form and into a centered form. We consider an equivalent formulation of the linear kinetic transport equation for neutral particles neutrons, photons as a system of two equations for even and odd parts of the distribution function. Citeseerx document details isaac councill, lee giles, pradeep teregowda. Even parity pn even parity sn ictt 2011 barbarino dulla ravetto mund ganapol 8 vladimirovs even parity formulation. Recently 5, 6, 7 a discretization in time scheme for the transient evenparity neutron transport equation was successfully developed and implemented in the variantkin3d code. This solver, named fiesta, is based on the secondorder even parity form of the transport equation.
A fast jacobianfree newtonkrylov iterative solver for. If the number of one bits adds up to an odd number, the parity bit is set to one. Even parity, or second order, neutron transport has been used in a limited capacity historically due to advantages and popularity of other deterministic methods. A number of test are examined via the code entrans, developed for even parity neutron transport. A new 2d3d transport core solver for the timeindependent boltzmann transport equation is presented. Monte carlo methods probability distribution functions.
An investigation was performed into the acceleration properties of the introduction of a nonlinear quasidiffusionlike tensor in linear and nonlinear solution schemes. It remains today, an important theoretical technique for investigating nonequilibrium. The theory of maximum principles is based on the cauchyschwartz inequality and the properties of a leakage operator g and a removal operator c. To formulate finite element methods variational y, the withingroup transport equation first is cast into the secondorder form that is even parity in angle. Received 21 apr 2018 revised manuscript received 23 oct 2018 accepted 31 oct 2018 published june 2019 r. Authors of this paper have already introduced the code entrans developed based on even parity neutron transport yousefi et al.
The general mathematical model of neutron transport is provided by the linear boltzmanns transport equation and the thesis begins with its precise mathematical formulation and presentation of known con ditions for its wellposedness. Using the tensor reduced matrix as a preconditioner for the conjugate gradients method proves highly efficient and effective. The variational formulation of the evenparity transport equation, originally. In this paper a method is presented for the application of energydependent spatial meshes applied to the multigroup, secondorder, even parity form of the neutron transport equation using isogeometric. Convergence acceleration methods for evenparity transport are being developed that have the potential to speed up transport calculations and provide a natural avenue for an implicitly coupled multiphysics code. To simplify the presentation, the transport equation is written in the even parity form.
Stabilized upwind petrovgalerkin supg and generalized least squares gls methods. The first part looks at basic reactor physics, including, but not limited to nuclear reactions, diffusion theory, reactor dynamics, fuel burnup and reactor safety. The coupled problem is analyzed theoretically and numerically. Even parity refers to a parity checking mode in asynchronous communication systems in which an extra bit, called a parity bit, is set to zero if there is an even number of one bits in a onebyte data item.
Adaptive refinement for pn neutron transport equation. A two dimensional multigroup, triangular mesh discrete ordinates explicit neutron transport. Finite element approximation to the evenparity transport equation. Nonlinear acceleration methods for even parity neutron transport 216 w. The object of this book is to present a balanced overview of the computational methods currently available for the solution of neutron transport problems encountered in engineering analysis. The angular discretization is performed through the expansion of the angular neutron. The boundary conditions of the spherical harmonics approximation are manipulated to have the forms. Finite element investigations for the construction of composite solutions of even parity transport equation anwar m. Convergence acceleration methods for evenparity transport were developed that have the potential to speed up transport calculations and provide a natural avenue for an implicitly coupled. The variational functional is constructed that reproduces the even parity neutron transport equation with isotropic scattering. Pdf neutron transport basics taught during a one semester course in. We consider an equivalent formulation of the linear kinetic transport equation for neutral particles neutrons, photons as a system of two. An ionization chamber for fission fragment analysis, drew mader. Although it has not been widely used, the numerical advantages for even parity transport are plenty, especially when.
The second part then deals with such physically and mathematically more advanced topics as neutron. In practical situations, fast iterative methods applied to improve the convergence order of the power iterations. In the algebraic form of the integral we clearly identify the net result of two opposite processes, i. Mathematical modeling of neutron transport milan hanu s department of mathematics university of west bohemia, pilsen thesis submitted in partial ful llment of the requirements for the degree of doctor of philosophy applied mathematics supervisor.
The spatial dependence of the even parity and odd parity angular flux has been modeled by discontinuous finite element method. Neutron transport theory nuclear reactor physics wiley. Nonlinear acceleration methods for evenparity neutron transport. We prove that the neutron density is factored in the product of two terms. By the adjoint weighted even parity flux, we can obtain the multigroup response fluxes over arbitrary shaped multidimensional geometries with less computational efforts compared to full parity approaches. Multidimensional shield performance analysis through an. A variational treatment of the finite element method for neutron transport is used based on a version of the even parity boltzman equation for the general case of anisotropic scattering and sources.
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