A finite element method for neutron transport. Part IV: A comparison of some finite element solutions of two group benchmark problems with conventional solutions

The one group finite element formulation of the neutron transport equation is adapted to treat multigroup shielding problems. The method is extended to eigenvalue problems by using a source iteration technique. The results for one-dimensional two group shielding problems show that the finite element method is fast and accurate; in the case of transport problems the solutions are free from the ray defects often found with discrete ordinate methods. Results for eigenvalue problems show that the method, when compared with discrete ordinate and collision probability methods and with diffusion theory in appropriate cases, is again fast and accurate. In the majority of cases, approximate values of the lowest eigenvalue when plotted against the reciprocal of the square number of nodes, lie very close to a straight line; consequently a very good estimate of the benchmark eigenvalue can often be found with coarse finite element methods. The results for two group problems have shown that the accuracy and speed achieved for the corresponding one group benchmark problems are maintained. These results and those of Part III, for the two-dimensional two region one group problems indicate that the finite element method is promising for the multigroup two dimensional problems. For both sets of results the finite element representation is used for the spatial dependence of the angular flux. The directional dependence of the angular flux is treated by expansions: either the Spherical Harmonics, the continuous representation; or Walsh Functions, the discrete representation. Walsh Functions do not appear to have any particular advantage over Spherical Harmonics. In the case of one dimension when the Spherical Harmonics reduce to Legendre functions, they are superior to Walsh Functions.     

AFEN method and its solutions of the hexagonal three-dimensional VVER-1000 benchmark problem

The unique features of the analytic function expansion nodal (AFEN) method in hexagonal-z geometry are described. The COREDAX code implementing the AFEN method is verified testing on the VVER-440 benchmark problem and a “simplified” VVER-1000 benchmark problem. The COREDAX code then applied to the original VVER-1000 benchmark problem exercise 2 (HZP case and HP case) provides very good results in comparison with those of other benchmark participants.     

Boundary conditions in the method of spherical harmonics

The equations of the method of spherical harmonics are derived for use with the single-velocity transport equation. From an analysis of the equations, conditions are obtained for the boundary between two different scattering media. It is shown that these conditions completely define a solution in any-order approximation (we are concerned here with the so-called P_N approximations) including even-order approximations. The possibility of using even-order approximations is of major practical interest. Thus, for example, the relatively simple P₂ approximation may serve as a method for improving the elementary diffusion theory without introducing excessive complexity.Even-order approximations, and in particular the p₂ approximation, have not as yet been used in calculations.In conclusion, the author wishes to thank Candidate of Physicomathematical Sciences, V. V. Orlov, for his consideration, interest and frequent discussions of the results. Thanks are also due to É. I. Gladyshev for assistance, and to all other colleagues who took part in discussions.     

Finite element analysis of mixed-mode thermoelastic fracture problems

This paper presents a rigorous hybrid finite element procedure for the analysis of the thermoelastic problems with mixed-mode cracks. The singular character of the temperature gradient which significantly affects the distribution of thermal stresses near the crack-tip is precisely described. The calculations of temperature and thermal stress fields are carried out by the finite element assemblage in which hybrid singular elements are used around the crack-tip and high-order isoparametric regular elements are taken elsewhere.To determine the mixed-mode stress intensity factors, both the direct extrapolation method based on the finite element solutions and the indirect method using the modified integrals accounting for thermal effects have been proposed. To avoid the underestimation of stress intensity factors obtained by quarter-point singular elements, the important role of the hybrid singular elements developed is also demonstrated.For verification purposes, several pure mode and mixed-mode problems are solved. Excellent correlations between the computer results and available referenced solutions are drawn.     

Finite element analysis of steady-state nonlinear heat transfer problems

A method of analysis and the associated computer program are presented for the purpose of solving steady-state nonlinear heat transfer problems in two-dimensional structures. The nonlinearity arises from the dependence of the thermal conductivities on temperature as well as from the presence of rediative heat transfer between parts of the structure. The problem is formulated in terms of an integral of conductivity and solved in an iterative way via the finite element concept. Several examples are given to illustrate the validity and practicality of the suggested solution technique.     

Finite element analysis of crack propagation in three-dimensional solids under cyclic loading

This paper is concerned with an efficient computational procedure for analyzing crack propagation in solids. The method is general; however, its application to semi-elliptical surface cracks in thick plates is discussed in particular. The strain energy release rate G for a crack in mode I is a function of the crack geometry, the direction of crack propagation and the state of loading. When G is known, the stress intensity factor K_I can easily be obtained. In this paper the strain energy of the plate is computed numerically for a wide range of crack geometries using the finite element method. A 20-node isoparametric solid element is employed in modelling the structure. Certain special techniques for increasing the computational efficiency of the method, such as multilevel subdivision of the structure (substructuring) and condensation of degrees of freedom that are not needed in the crack propagation analysis, are emphasized. In fact, analysis of a large number of crack geometries requires only insignificantly more computational efforts than treating a single crack. Certain other aspects of the finite element modelling are also discussed.Two methods for replacing the computed discrete values of strain energy by continuous functions are presented. These functions are expressed in terms of the two half-axes defining the geometry of the elliptical crack and they are determined using a least square technique. G and K_I are easily deduced from these functions. As an example, a semi-elliptical, part-through, surface crack in a thick nickel steel plate is analyzed. The crack is subjected to a combination of axial and bending loading, applied cyclically. From the finite element calculations of the strain energy and the stress intensity factors which are computed accordingly, crack propagation along the two half-axes of the ellipse is calculated by utilization of a formula suggested by Paris. The results are checked against laboratory fatigue tests. The method has proved to be very efficient and accurate, and due to its generality it can also be applied to complicated geometries and complex states of loading.     

An exact general solution in spherical harmonics of the Boltzmann equation

A solution of the one-velocity kinetic Boltzmann equation is obtained as a series of spherical harmonics. General expressions are obtained for the terms of the series, derived without any approximately valid assumptions. As particular cases of this solution, we obtain formulas for the known P_N-approximations for the spherical-harmonic method.The exact general solution of the kinetic equation in the form of a series of spherical harmonics contains arbitrary functions which must depend on the formulation of boundary conditions. The general determination of the boundary conditions and the arbitrary functions is not considered. All the results of [4] remain valid for P_N-approximations.Translated from Atomnaya Énergiya, Vol. 18, No. 5, pp. 459–463, May, 1965     

Finite element analysis of nonlinear problems in the dynamical theory of coupled thermoelasticity

This paper is concerned with the formulation of finite element models for the analysis of nonlinear problems in the dynamical theory of coupled thermoelasticity. Displacement and temperature fields are approximated over a finite element, and energy balances are formed which lead to general equations of heat conduction and motion of thermoelastic finite elements. A number of special cases are considered. Numerical results obtained from simple representative problems are also presented.     

A spherical harmonics—Finite element discretization of the self-adjoint angular flux neutron transport equation

The spherical harmonics (P_N) method is widely used in solving the neutron transport equation, but it has some disadvantages. One of them comes from the complexity of the P_N equations. Another one comes from the difficulty of dealing with the vacuum boundary condition exactly. In this paper, the P_N method is applied to the self-adjoint angular flux (SAAF) neutron transport equation and a set of P_N moments equations coupled with each other are obtained. An iterative method is utilized to decouple them and solve them moment by moment. The corresponding vacuum boundary condition is derived based on the Marshak boundary condition. The spatial variables are discretized on unstructured-meshes by use of the finite element method (FEM). The numerical results of several problems demonstrate that this method can provide high precision results and avoid the ray effect, which appears in the discrete ordinate (S_N) method, with relatively high computational efficiency.     

Use of the surface harmonics method forcalculation of 2D C5G7 MOX benchmark

The C5G7 MOX benchmark specifying a sixteen-assembly core with asurrounding water reflector was proposed as a basis to measure current transport code abilities in the treatment of reactor core problems without spatial homogenization. Seven-group cross sections for all materials were used as initial information. Just that fact allows to test an accuracy of solving the neutron transport equation excluding additional errors connected with preparing the group cross sections. In this paper, Surface Harmonics Method (SHM) is applied to calculation of the two-dimensional configuration of this benchmark. Different approximations of SHM were applied, both with and without spatial homogenization. Additionally, this fact allowed evaluating the effect of spatial homogenization of cells. Comparisons were carried out for k_eff and pin powers both with the reference results and between the results calculated by different SHM approximations.     

Elasto-plastic analysis of three-dimensional structures using the isoparametric element

A method for elasto-plastic analysis of three-dimensional structures is presented. The finite element method, implemented by a three-dimensional isoparametric element is used in the study. An “initial” stress approach is employed in the range of plastic behavior.     

Finite element methods in the analysis of reactor vessels

The application of the finite element method to linear and non-linear problems in pressure vessel technology is presented. New developments for dealing with components such as liners, prestressing cables and reinforcement are outlined and some improvements possible in thin shell situations are discussed. A general solution technique for non-linear analysis is presented and applied firstly to the problem of the plastic behaviour of steel pressure vessels. The failure of PCRVs by concrete cracking is then considered. Finally, the time-dependent phenomenon of creep is discussed. In all cases the theory is illustrated by practical examples.     

On the finite element solution of the three-dimensional tire contact problem

Starting from a theoretical analysis of the static contact problem “deformable body-rigid object” various solutions including the three-dimensional pneumatic tire contact problem are being discussed. The computed shapes and sizes of the footprint area as well as the load-deflection response are in good agreement with experimental results. The following approaches for a numerical solution are presented in this paper:

1. (1) Elimination method: This method is based on using the influence coefficient technique to compute the nodal forces as well as the footprint area by iteration applying a geometrical linear or nonlinear FE technique.

2. (2) ‘Nonlinear Programming’ technique: Recent developments in the theory of nonlinear optimization in abstract spaces suggest to describe the contact problem by a constrained optimization problem in function spaces. The minimization of the potential energy of the elastic body, the rigid obstacle being replaced by an operator constraint depending on the displacement functions, leads to three necessary conditions determining completely the equations of equilibrium, the contact zone and the pressure distributions in the contact zone. This result justifies the use of a nonlinear programming code to solve the discrete problem approximated by the method of finite elements.

3. (3) Variation of boundary conditions: This method is based on an incremental formulation including nonlinear geometry and variable boundary conditions.

Computational model of the evolution of the stress-strain state of a spherical fuel element

The problem of a spherically symmetric stress state in a spherical fuel element with a cavity at the center for collecting gaseous fission products is examined. A system of equations, which are exact within the framework of the physical model, is derived for this case, a numerical method for solving this system is proposed, and the evolution of the strain in the cladding is studied for characteristic cases. The results of numerical estimates of the temporal development of strain in the cladding are presented. 2 figures. 10 references. All-Russia Scientific-Research Institute of Automatic Machine Engineering. Translated from Atomnaya énergiya, vol. 88, No. 2, pp. 137–142, February, 2000.     

Effect of the parameters of a three-dimensional cylindrical radiator and a spherical detector on measurement results

The correction factors for spherical and hollow spherical detectors for point and cylindrical radiators arranged with the generatrix toward the detector are calculated. For a spherical detector the correction factor is always greater than or equal to 1, and the maximum value does not exceed 2 for a point radiator. For a cylindrical three-dimensional radiator for h≥2E_max the correction factor reaches the maximum value when h=2E_max and R→0. When the detector is positioned flush against a cylindrical radiator, its correction factor (for large values of h/R) is less than for a point radiator placed at the geometric center. As this ratio decreases, the correction factor for a cylindrical radiator becomes greater than the corresponding factor for a point radiator. As the self-absorption of γ rays in the material of a cylindrical radiator increases, the correction factor decreases. A large difference is not observed between the correction factors obtained with and without the scattered component of the radiation in the source material. 2 figures, 4 tables, 5 references. Northwest Civil Service Academy. Translated from Atomnaya énergiya, Vol. 86, No. 1, pp. 32–36, January, 1999.     

A parametric analysis of creep buckling of a shallow spherical shell by the finite element method

This paper presents a parametric analysis of the creep buckling of a shallow spherical shell subjected to uniform external pressure using the finite-element incremental method. Axisymmetric creep deformation analysis is first performed to obtain pre-buckling behavior by the time incremental method. The buckling modes considered are both bifurcation and axisymmetric snap-through types. Fourier expansion technique is conveniently employed to construct the stiffness matrices corresponding to asymmetric bifurcation as well as axisymmetric snap-through modes of deformation. The critical time for creep buckling is defined by the time at which either stiffness matrices corresponding to the axisymmetric or asymmetric modes lose their positive definiteness. The critical times and the buckling modes are obtained over a fairly wide range of the geometric parameters pertaining to shallow spherical shell subjected to uniform external pressure.