A variational treatment for the time dependent boltzmann equation as a basis for numerical solutions conserving neutrons

A maximum principle for the time-dependent first-order Boltzmann equation is established in two independent ways:- by a generalized least squares method and by a method based on the properties of an appropriate bi-linear form. The second derivation suggests a metric for a Hilbert space which provides a geometrical interpretation of the variational principle. This interpretation leads to a Petrov-Galerkin weighted residual method in contrast to the Galerkin method of Martin for time dependent transport.

The maximum principle is used to define a figure of merit for the global error of any numerical solution for time dependent transport. The principle is used also to demonstrate the neutron conservation property of optimized numerical solutions, and the convergence of finite element methods based on the variational principle.

A direct use of the maximum principle give a conservative three level scheme for transients in the angular flux. A less massive calculation for a conservative solution proceeds in two stages. The first stage obtains a preliminary solution based on a sequence of tailored steady state calculations for the even-parity angular flux. The relevant equations are derived by making residual terms vanish in the variational principle. They are equivalent to the finite element - finite difference equations used by de Oliveira and Wood for the analysis of oil-well logging by means of a neutron pulse. For the second stage the preliminary solution is weighted at each time step. The weights are determined by the maximum principle to yield a conservative solution which is continuous in time. The solution can be arranged to have also a continuous time derivative. Thus the maximum principle can be used both as a way of predicting and correcting solutions.     

Duality in transport theory

The relationship between the forward and the adjoint or backward linear particle transport equations is treated from a different viewpoint. Both deterministic and stochastic transport equations are considered. The starting point is that at the level of the Green's function, the forward and backward transport equations represent a dual alternative for calculating the same quantity, and a close analogy with the similar properties of the forward and backward Kolmogorov or master equations of discrete Markovian stochastic processes is stressed. This analogy is exploited in that derivation methods of such master equations are transferred to transport equations. It is shown that forward and backward (adjoint) transport equations can be derived directly from the Markovian property of the transport process, without explicitly resorting to adjoint techniques. This principle is referred to as duality in this paper. The first application is a new derivation of the conventional forward and adjoint transport equations from one common balance equation, expressing the Markovian property of the transport process. Then forward- and backward-type transport master equations are derived, and it is shown that the corresponding transport equations are their first moments. Reasons for the rare occurrence of the forward master transport equation are discussed. From the master equations, second-order forward and backward moment equations are also derived. It is seen that for the derivation of forward-type equations for the doublet density, the forward two-point master equations are an especially convenient tool when external sources are involved.     

The role of the Boltzmann transport equation in radiation damage calculations

The purpose of this article is to discuss in some detail the uses of the Boltzmann equation in assessing displacement damage due to fast atoms in solids. We derive the appropriate equations for describing atomic motion in solids directly from the non-linear Boltzmann equation for mixtures. The basic assumption is that binary collisions are valid and that the solid can be regarded as amorphous or as a random lattice of atoms. Moreover, we assume that the density of fast atoms (i.e. those with energy greater than about 25 eV) is so small that they do not interact with each other. With these assumptions a set of linear, coupled Boltzmann transport equations are derived which describe the projectile and recoil atom distributions in space, time and velocity. Additionally, we show how electronic interactions may be accounted for in an accurate and analytically useful manner.The Boltzmann equations thus derived are in the so-called forward form, that is the final co-ordinates are the ones operated upon rather than the initial ones. We consider the corresponding adjoint equations and from the mathematical properties of the operators prove a reciprocity relation between the solutions of the forward equations and the adjoint ones. This enables either equation, forward or backward (adjoint), to be used to calculate the distribution function, a fact which is extremely useful in simplifying calculations for certain types of problem. It also demonstrates the inter-relationships between the works of various groups.The essentials of the scattering cross-sections are discussed and it is seen that two distinct problems arise: scattering in the C.M. system, which is determined by the interparticle force law, and scattering in the L-system which is governed only by the laws of conservation of energy and momentum. In both cases we discuss analytical simplifications which lead to useful approximations for solving the transport equation in closed form.One of the basic problems of radiation damage theory is that of sputtering and we spend some time explaining how binding forces at a vacuum-solid interface affect the solution of the transport equation.Several infinite medium problems are solved using backward and forward equations, thus we calculate the collision density of recoiling atoms due to a high energy source and also the time dependent behaviour of the energy spectrum following a pulse of fast atoms. The energy deposition and total number of moving particles at time t is calculated for various scattering models. The importance of electronic stopping in calculating the amount of energy eventually ending up in atomic motion is assessed via a time dependent forward equation. Its connection with the work of Lindhard is pointed out.We discuss displacement damage and show how the number of displacements per primary particle may be calculated via the collision density approach or through the backward equation with suitably modified energy transfer terms. The ideas of energy partitioning are discussed and methods for assessing its accuracy outlined.The spatial distribution of radiation damage receives considerable effort and we look at the various methods of dealing with the anisotropy of scattering, from Legendre polynomial expansions to the straight-ahead, or path-length, approximation. Analytical solutions are given for a variety of simple problems and in particular the vexed question of ion implantation in heterogeneous layered structures is discussed and various solutions proposed. For infinite, homogeneous media we discuss the method of moments using forward and backward equations and conclude that in many situations the forward form has much to commend it although in some situations the backward equation is not without merit. They are indeed complementary techniques.Sigmund's method for calculating the sputtering yield and efficiency is described via the backward equation and the approximations introduced by his “infinite medium boundary condition” are examined. We propose a more accurate theory which includes the half space nature of the system but which, because of this, leads to difficult mathematical problems. Certain simplified problems are solved and ideas for future progress are put forward.The concept of channelling is discussed and methods for including it directly into the Boltzmann equation are outlined. Essentially, these suggest the introduction of an angularly dependent cross-section which accounts in a phenomenological manner for the anisotropy of the microstructure of the solid.A final section is devoted to a stochastic formulation of the particle distribution function through a probability density. An equation is obtained for this density by probability balance and is then simplified by the introduction of a generating function. Successive differentiations of the generating function enable equations for the average value to be obtained, i.e. the conventional Boltzmann equation, and also higher order correlation functions which give a measure of the fluctuations in the number of particles in a cascade. Equations for vacancies and interstitials and their correlations are discussed in this respect.Finally, it should be noted that this paper shows a very personal and possibly biased view of radiation damage calculation. Indeed, it is not intended to discuss radiation damage as such, but rather to outline the difficulties and particularly to show how the Boltzmann equation has beeb and can be used in its forward and backward forms to understand atomic displacement problems.     

Completely boundary-free minimum and maximum principles for neutron transport and their least-squares and Galerkin equivalents

Some minimum and maximum variational principles for even-parity neutron transport are reviewed and the corresponding principles for odd-parity transport are derived by a simple method to show why the essential boundary conditions associated with these maximum principles have to be imposed. The method also shows why both the essential and some of the natural boundary conditions associated with these minimum principles have to be imposed. These imposed boundary conditions for trial functions in the variational principles limit the choice of the finite element used to represent trial functions. The reasons for the boundary conditions imposed on the principles for even- and odd-parity transport point the way to a treatment of composite neutron transport, for which completely boundary-free maximum and minimum principles are derived from a functional identity. In general a trial function is used for each parity in the composite neutron transport, but this can be reduced to one without any boundary conditions having to be imposed.An alternative derivation of the functional identity gives as a by-product Davis complementary principles for composite neutron transport, which use two trial functions satisfying essential boundary conditions. If these two trial functions are replaced by one then both natural and essential boundary conditions have to be imposed.The functional identity is used to re-establish three well-known principles directly, and it shows that the boundary-free maximum principle is equivalent to a generalized least-squares method with weights in the form of operators and no boundary conditions imposed.The least-squares principle uses two positive definite volume integrals so that the divergence therorem can be used to change awkward volume integrals into manageable surface integrals without the need to impose boundary conditions on trial functions.A geometrical interpretation of the boundary-free maximum principle is given using the projection theorem for a Hilbert space with a suitable metric. This path leads to several boundary-free Galerkin equations for both the second-order and first-order forms of the transport equation.     

Comparison of finite-difference and variational solutions to advection-diffusion problems

Two numerical solution methods are developed for 1-D time-dependent advection-diffusion problems on infinite and finite domains. Numerical solutions are compared with analytical results for constant coefficients and various boundary conditions. A finite-difference spectrum method is solved exactly in time for periodic boundary conditions by a matrix operator method and exhibits excellent accuracy compared with other methods, especially at late times, where it is also computationally more efficient. Finite-system solutions are determined from a conservational variational principle with cubic spatial trial functions and solved in time by a matrix operator method. Comparisons of problems with few nodes show excellent agreement with analytical solutions and exhibit the necessity of implementing Lagrangian conservational constraints for physically-correct solutions.     

A finite element formulation for thermal stress analysis. Part II: Finite element formulation

This paper deals with the development of a variational principle which can be used for solving problems related to the thermoelastic behavior of solids and is the first of the two part series. The formulation is based on the introduction of a new quantity defined as heat displacement and related to temperature in the same manner as the mechanical displacement is related to strain. The introduction of such a quantity allows the heat conduction equations to be written in a form equivalent to the equation of motion, and the equations of coupled thermoelasticity to be written in a unified form. The obtained equations are used to write a variational formulation which, together with the concept of generalized coordinates, yield a set of differential equations with the time as the independent variable. These equations can be used to formulate a finite element solution for thermoelastic problems. This is done in the second part.     

A finite element method for neutron transport—I. Some theoretical considerations

A variational treatment of the finite element method for neutron transport is given based on a version of the even-parity Boltzmann equation which does not assume that the differential scattering cross-section has a spherical harmonic expansion. The theory of minimum and maximum principles is based on the Cauchy-Schwartz inequality and the properties of a leakage operator G and a removal operator C.For systems with extraneous sources, two maximum and one minimum principles are given in boundary free form, to ease finite element computations. The global error of an approximate variational solution is given, the relationship of one of the maximum principles of the method of least squares is shown, and the way in which approximate solutions converge locally to the exact solution is established. A method for constructing local error bounds is given, based on the connection between the variational method and the method of the hypercircle.The source iteration technique and a maximum principle for a system with extraneous sources suggests a functional for a variational principle for a self-sustaining system. The principle gives, as a consequence of the properties of G and C, an upper bound to the lowest eigenvalue. A related functional can be used to determine both upper and lower bounds for the lowest eigenvalue from an inspection of any approximate solution for the lowest eigenfunction.The basis for the finite element method is presented in a general form so that two modes of exploitation can be undertaken readily. The finite element model can be in phase space, with positional and directional co-ordinates defining points of the model, or the finite element model can be restricted to the positional co-ordinates and an expansion in orthogonal functions used for the directional co-ordinates. Suitable sets of functions are spherical harmonics and Walsh functions. The latter set is appropriate if a discrete direction representation of the angular flux is required.     

A perturbation theory for the non-linear system of neutron transport and burnup equations

A perturbation theory for use in nuclear reactor burnup analysis is derived. An important characteristic function is defined, and the related adjoint equation is obtained simply by using the variational principle. The adjoint matrix operator is evaluated directly from its defining differential expression. Responses at the end of cycle due to changes in initial material inventory, in nuclear data, and in power demands can be calculated using the previously determined forward and adjoint solutions. The method has additional applications, notably in selecting beginning of cycle conditions so as to achieve a particular end of cycle condition.     

Extremum principles in the dynamics of rigid-plastic bodies: A critical review of existing applications

In recent years a number of variational or extremum principles have been formulated for bodies deforming beyond the elastic range as a result of dynamic loading. These theorems represent a potential for deriving approximate solutions to the initial-boundary value problems for inelastic continua and structures. However, some of the principle are of much more restrictive character than it is commonly believed and care should be taken in interpreting properly the resulting approximations. The importance of applying a variational technique in deriving approximate solutions has somehow been overemphasized in the literature and some authors tried to demonstrate advantages of the suggested methods while no mention was made of the existing shortcomings. This article aims to discuss some of the limitations and clarify difficulties in applying the extremum principles to dynamic problems for rigid-plastic continua and structures with special reference to the development of analytical methods. In particular, an attempt is to answer the following questions: (1) To what extent the existing methods can be used in the analysis of transient problems for impulsively or pulse loaded structures? (2) Under what conditions stationarity of appropriate functionals can be proved so that use of direct methods of the calculus of variations is well legitimated? (3) Can a formal approximation method be developed for mode form response in which exact solution is approached with any desired degree of accuracy?These questions are related to the problem of a choice of a class of admissible functions. If the admissible functions contain the true solution, then the extremal theorems are shown to be applicable for deriving solutions to both transient and fixed shape problems. However, such a situation can be regarded as exceptional, and in typical cases the exact solution is either unknown or cannot be expressed in terms of elementary functions. It is pointed out that in these circumstances solutions to transient problems, obtained by means of analytical methods, might not be correct. Examples from the available literature are cited. In contrast, approximate mode form solutions can still be derived from variational principles, but then the question of accuracy arises. Most of the existing solutions were obtained from the kinematic principle by considering very simple admissible fields of velocities or accelerations involving only a few free parameters. Because of the non-linearity of the dissipation function and lack of the property of superposition it becomes very tedious or even impossible to evaluate effectively integrals appearing in the functionals when more terms are considered. Thus, the Rayleigh-Ritz method for finding a stationary point of a functional does not seem to be of great help when an improved accuracy is desired. Two ways of overcoming this difficulty are suggested. In the first, possibilities of using a dynamic rather than kinematic principle is explored, the former being expressed entirely in stresses. As an alternative, a new definition of a set of approximate functions is introduced and its advantages over Ritz's coordinate functions are explained on the plate problem. Finally, conditions are examined for the extrema to be stationary. Various types of extremal behaviour with analytic and non-analytic extrema are demonstrated, depending on the smootheness of the yield conditions and continuity in the slope of the admissible velocity field.     

A least squares principle unifying finite element,finite difference and nodal methods for diffusion theory

A least squares principle is described which uses a penalty function treatment of boundary and interface conditions. Appropriate choices of the trial functions and vectors emplyoed in a dual representation of an approximate solution established complementary principles for the diffusion equation. A geometrical interpretation of the principles provides weighted residual methods for diffusion theory, thus establishing a unification of least squares, variational and weighted residual methods.The complementary principles are used with either a trial function for the flux or a trial vector for the current to establish for regular meshes a connection between finite element, finite difference and nodal methods, which can be exact if the mesh pitches are chosen appropriately. Whereas the coefficients in the usual nodal equations have to be determined iteratively, those derived via the complementary principles are given explicitly in terms of the data.For the further development of the connection between finite element, finite difference and nodal methods, some hybrid variational methods are described which employ both a trial function and a trial vector.     

Convergence problems associated with the iteration of adjoint equations in nuclear reactor theory

Convergence problems associated with the iteration of adjoint equations based on two-group neutron diffusion theory approximations in slab geometry are considered. For this purpose first-order variational techniques are adopted to minimise numerical errors involved. The importance of deriving the adjoint source from a breeding ratio is illustrated. The results obtained are consistent with the expected improvement in accuracy.     

Comparison of analytical and variational solutions of steady-state neutron-diffusion problems

Classical variational techniques are used to obtain accurate solutions to the multigroup multiregion 1-D steady-state neutron-diffusion equation. Analytic solutions are constructed for benchmark verification. Functionals with cubic trial functions and conservational Lagrangian constraints are developed and compared with non-conservational functionals. Eigenvalues, neutron balance and relative flux and current values at interfaces are compared to exact analytical results. Excellent agreement is obtained with conservational functionals using cubic trial functions in comparisons with analytic solutions.     

Methods for Solving Adjoint Equations in Nonlinear Conditionally-Critical Problems

The special features of methods for calculating the neutron value functions with respect to functionals which are determined by solving a nonlinear conditionally-critical neutron transport equation are examined. The adjoint equations for these functions are written in an abstract operator form and contain additional, compared with linear problems, terms which are orthogonal to the neutron flux. These terms take account of the contribution due to the change in the properties of the medium when neutrons are introduced into the reactor into the balance of the values. The operators of the adjoint problem are analyzed and criteria ensuring that the proposed iteration methods converge are formulated. The satisfaction of the criteria is checked for the solution of problems where the nonlinearity of the emission equations is due to the dependence of the concentration of fuel nuclei on the neutron flux. It is noted that the iteration processes converge rapidly.     

Propagation noise calculations in VVER-type reactor core

Neutron noise induced by propagating disturbances in VVER-type reactor core is addressed in this paper. The spatial discretization of the governing equations is based on the box-scheme finite difference method for triangular-z geometry. Using the derived equations, a 3-D 2-group neutron noise simulator (called TRIDYN-3) is developed for hexagonal-structured reactor core, by which the discrete form of both the forward and adjoint reactor dynamic transfer functions (in the frequency domain) can be calculated. In addition, both types of noise sources, namely point-like and traveling perturbations, can be modeled by TRIDYN-3. The results are then benchmarked in different cases. Considering the noise source as propagating perturbations of the macroscopic absorption cross sections, the induced neutron noise is calculated throughout the reactor core. For the first time, adjoint approach is applied and examined for modeling moving noise sources. Moreover, the space- and frequency-dependence of the propagation noise are investigated in this paper.     

A sensitivity analysis of thermal lattices kinetic parameters with respect to the spectral weighting function using ultrafine BN method

Accurate calculation of kinetic parameters is of utmost importance in the safety analysis of a nuclear reactor. In the current paper, two approaches are investigated to evaluate these parameters in energy phase space. In the first approach, these parameters are derived from an energy-continuous form of the forward and adjoint transport equations and then integrals with respect to the energy variable are replaced by weighted summations over the energy groups, while in the second approach these parameters are extracted from the multi-group forward equation and its associate adjoint equation in which their multigroup constants are weighted by forward spectrum. The difference of weighting functions in these two approaches would naturally lead to different values for the kinetic parameters. This paper mainly compares the outcome of these two approaches in calculating kinetic parameters for two main types of thermal critical lattices: Mixed Oxide (MOX) and Uranium Oxide (UOX) using ultrafine B_N method. The results show that calculations which are based on using the forward weighted spectrum for generating the kinetic parameters underestimate prompt neutron generation time in both thermal lattices, while effective delayed neutron fraction is overestimated in UOX thermal lattice and underestimated in MOX one.     

A multigroup energy formalism for reactor stochastic equations in the space-independent low power model

The time-dependent probability distribution of neutrons in a space-independent, low-power, multiplying assembly with a source is developed in the multigroup energy approximation as forward and backward Kolmogorov equations. The relationship between these as adjoint equations is made explicit in a tensor notation and the equations developed in the generating function formalism.     

Second-order functionals for spectral synthesis with discontinuous trial functions

As an extension of earlier work, the most general second-order variational functional for the two-region multigroup neutron diffusion problem has been derived from the Generalized Roussopoulos Principle. Several special cases of this general functional have been identified with second-order functionals previously proposed in the literature. A theorem on the equivalence of the first- and second-order synthesis has been proved under certain assumptions. This theorem provides a criterion for discriminating between the pure first-order and pure second-order cases. It is evident that the second-order synthesis is advantageous over the first-order, because one has to choose fewer trial functions. Some numerical results, based on the pure first- and pure second-order synthesis, for a prototype two-region two-group reactor model have been given. For the purpose of illustration, two special cases, viz. Galerkin weighting and unity weighting, for the choice of adjoint trial functions for the second-order synthesis have been considered.     

A multi group multiparticle formalism for multiplying systems Kolmogorov equations in the space-independent low-power model

The time-dependent probability distribution of neutrons, precursors and detectrons in a space-independent, low-power, multiplying assembly with a source is developed in the multigroup approximation as forward and backward Kolmogorov equations. The relationship between these as adjoint equations is made explicit in a tensor notation and the equations developed in the generating-function formalism.     

Analytic solutions to mass transport equations for cylindrical nuclear fuel elements

The transport equation characterizing pore movement has several analytic solutions depending on the form of the pore velocity, υ. Properties of this equation are examined with special reference to boundary conditions, movement of turning points where the porosity is stationary, and the possibility of formation of rings of increased porosity. Solutions of the equation with a source term are also obtained. The equations for solid state mass transfer in temperature gradients, where the motion is diffusive, have similar asymptotic solutions for sufficiently small times whose explicit forms are obtained as functions of the diffusivity and heat of transport. These analytic solutions enable simple estimations to be made of the time necessary to redistribute material out to a given radius in terms of the basic parameters.