Monte carlo methods probability distribution functions. Park acceleration technique using krylov subspace methods fo r 2d arbitrary geometry characteristics. 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. New insights into numerical solutions of the even transport equation in twodimensional xy geometry noh, taewan. Finite element investigations for the construction of. Transport approximations the spectral element method is well suited for the second order form of the transport equation. A conjoint variational formulation based on discontinuous finite elements approach for pn neutron transport equation has been presented. With changes only in the interaction laws contained in the cross.
Authors of this paper have already introduced the code entrans developed based on even parity neutron transport yousefi et al. If the number of one bits adds up to an odd number, the parity bit is set to one. The code is able to solve multidimensional forward and adjoint neutron transport equation over an arbitrary geometry using linear or high order elements. The theory of maximum principles is based on the cauchyschwartz inequality and the properties of a leakage operator g and a removal operator c. The particle scattering integral of even and odd parity transport equations is converted into a nonlinear algebraic form and into a centered form. A fast jacobianfree newtonkrylov iterative solver for.
To simplify the presentation, the transport equation is written in the even parity form. The withingroup even parity neutron transport equation is formulated with complex angular and spatial trial functions and with a complex buckling approximation. Once moved into the nonlinear solution scheme, the implicit coupling of the convergence accelerated transport method into codes for. The coupled problem is analyzed theoretically and numerically. The second order even parity form of the transport equation is discretized using the continuous galerkin finite element method in. The even parity equation is solved on the fine mesh using moc, while the odd parity.
The angular discretization is performed through the expansion of the angular neutron. Neutron transport equations, newtonkrylovschwarz, mesh partitioning, workload balancing, parallel computation, multilevel domain decomposition 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. Nonlinear acceleration methods for evenparity neutron transport.
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. Exppg yloiting concurrency at the petascale in neutron. It is a nonlinear integrodifferential equation for the phase space density of the molecules of a dilute gas. Although it has not been widely used, the numerical advantages for even parity transport are plenty, especially when. Nonlinear acceleration methods for even parity neutron transport, william martin. Computational methods of neutron transport book osti. Development of code, pnfent, based on using finite. A goaloriented and selfadaptive mesh refinement approach. This solver, named fiesta, is based on the secondorder even parity form of the transport equation. Nonlinear acceleration methods for evenparity neutron.
Multidimensional shield performance analysis through an. A new 2d3d transport core solver for the timeindependent boltzmann transport equation is presented. 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. It is a solution of a heterogeneous transport problem with. The spherical harmonics approximation is based upon the second order even parity form of the neutron transport equation. The second part then deals with such physically and mathematically more advanced topics as neutron. Citeseerx document details isaac councill, lee giles, pradeep teregowda. Fpga hardware acceleration for high performance neutron transport. Finite element investigations for the construction of composite solutions of even parity transport equation anwar m. A modular nodal method is developed for solving the neutron transport equation by using the spherical harmonics approximation in two dimensional cartesian coordinates. The particle scattering integral of even and oddparity transport equations is converted into a nonlinear algebraic form and into a centered form. Using the tensor reduced matrix as a preconditioner for the conjugate gradients method proves highly efficient and effective.
When a nucleus contains an odd number of both particle types, it is nearly always unstable. Received 21 apr 2018 revised manuscript received 23 oct 2018 accepted 31 oct 2018 published june 2019 r. In practical situations, fast iterative methods applied to improve the convergence order of the power iterations. 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. Although this angular dependency can be approximated by various approaches, the spherical harmonics approach offers some advantages. Stabilized upwind petrovgalerkin supg and generalized least squares gls methods. Coarse mesh methods for the transport calculation in. An improved variational nodal method for the solution of. A novel neutron recoil spectrometer concept utilizing heavyion recoils and time and spatiallyresolved sensor, joel long. Nodal diffusion and transport methods are formulated variationally in terms of the evenparity form of the neutron transport equation and. Discrete ordinates in 2d and 3d geometries with fe mixed mesh capabilities these second order methods have been implemented on. A two dimensional multigroup, triangular mesh discrete ordinates explicit neutron transport. International conference on the physics of reactors 2010. Starting from this result we have first more thoroughly investigated the order of approximation, with respect to the time variable, arising.
Even parity, or second order, neutron transport has been used in a limited capacity historically due to advantages and popularity of other deterministic methods. 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. The methods for the numerical solution of the neutron transport equation can be divided. In the algebraic form of the integral we clearly identify the net result of two opposite processes, i. The treatment of the neutron transport equation by the evenodd parity formalism. This third, completely revised edition of the textbook retains the proven concept of complete and balanced coverage of the topic. 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. Evenparity, or second order, neutron transport has been used in a limited capacity historically due to advantages and popularity of other deterministic methods. Read even and odd parity kinetic equations of particle transport. 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. 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. It remains today, an important theoretical technique for investigating nonequilibrium. The first part of the book covers basic reactor physics, including, but not limited to nuclear reaction data, neutron diffusion theory, reactor.
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. A modular nodal method for solving the neutron transport. Even and oddparity kinetic equations of particle transport. 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. 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. Pdf neutron transport basics taught during a one semester course in. The first one describes local behavior of the density at the cell level. Spherical harmonic method in 1d, 2d and 3d geometries with fe mixed mesh capabilities sn2nd. While it is possible to replace the even parity methodology in proteussn with a supg or gls scheme, this does not. Even parity pn even parity sn ictt 2011 barbarino dulla ravetto mund ganapol 8 vladimirovs even parity formulation. Nonlinear acceleration methods for even parity neutron transport 216 w. The principal application is to the deeppenetration transport of neutrons andor photons. 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.
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. 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. The spatial dependence of the even parity and odd parity angular flux has been modeled by discontinuous finite element method. Coupling of transport and diffusion models in linear. We consider an equivalent formulation of the linear kinetic transport equation for neutral particles neutrons, photons as a system of two.
Finite element investigations have been carried out to construct composite solutions of transport problems. An investigation was performed into the acceleration properties of the introduction of a nonlinear quasidiffusionlike tensor in linear and nonlinear solution schemes. 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. Nuclear engineering etds engineering etds university. The variational functional is constructed that reproduces the even parity neutron transport equation with isotropic scattering. Numerical solutions to neutron transport equation are required in reactor core. If the even parity form of the transport equation is used, the spherical harmonics approach provides a set of second order differential equations. 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. The second, developed here uses a variational principle as a point of departure for the application of finite elements to neutron transport problems. Finite element approximation to the evenparity transport equation. Although it has not been widely used, the numerical advantages for evenparity transport are plenty, especially when. Neutron transport theory nuclear reactor physics wiley. Adaptive refinement for pn neutron transport equation. Abstracta variational finite elementspherical harmonics method is presented for the solution of the even parity multigroup equations with anisotropic scattering and sources.
To formulate finite element methods variational y, the withingroup transport equation first is cast into the secondorder form that is even parity in angle. An ionization chamber for fission fragment analysis, drew mader. Evenparity neutron transport equation eigenvalue search nonlinear systems 1 introduction the keigenvalue calculation in criticality problems has traditionallyutilizedtheclassicalpoweriterationmethod which has slow convergence order. Coupled neutronics thermal hydraulics event thermix. Numerical methods in the theory of neutron transport book. Siam journal on mathematical analysis siam society for. 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. Multilevel acceleration of neutron transport calculations approved by. Proteussn is a threedimensional deterministic neutron transport code which solves the second order formulation of the neutron transport equation. We prove that the neutron density is factored in the product of two terms. A number of test are examined via the code entrans, developed for even parity neutron transport. 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. The variational formulation of the evenparity transport equation, originally.
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