Boundary Element Methods Applied to Two-Dimensional Neutron Diffusion Problems

The Boundary element method (BEM) has been applied to two-dimensional neutron diffusion problems. The boundary integral equation and its discretized form have been derived. Some numerical techniques have been developed, which can be applied to critical and fixed- source problems including multi-region ones. Two types of test programs have been developed according to whether the ‘zero-determinant search’ or the ‘source iteration’ technique is adopted for criticality search. Both programs require only the fluxes and currents on boundaries as the unknown variables. The former allows a reduction in computing time and memory in comparison with the finite element method (FEM). The latter is not always efficient in terms of computing time due to the domain integral related to the inhomogeneous source term; however, this domain integral can be replaced by the equivalent boundary integral for a region with a non-multiplying medium or with a uniform source, resulting in a significant reduction in computing time.

The BEM, as well as the FEM, is well suited for solving irregular geometrical problems for which the finite difference method (FDM) is unsuited. The BEM also solves problems with infinite domains, which cannot be solved by the ordinary FEM and FDM. Some simple test calculations are made to compare the BEM with the FEM and FDM, and discussions are made concerning the relative merits of the BEM and problems requiring future solution.     

On boundary element discretization of transient thermoelastic problems

A boundary element discretization scheme is presented for the transient thermoelastic problems. A set of integral equations on the boundary are derived from a weighted residual statement of the problem in which the fundamental solution is used as the weight function. The resulting boundary integral equation for the temperature field is discretized by means of the boundary element in space and time. It is then shown that the displacement field can be obtained under the same boundary element discretization.     

Boundary element method in nonlinear transient heat transfer of reactor solids with convection and radiation on surfaces

This paper deals with the boundary (integral) element method for non-steady conduction problems of solids, subject to non-linear convective and radiation conditions on surfaces. Boundary integral equations for the mixed-type and non-linear boundary conditions, both for the case with constant and variable heat conductivity are derived, modelled by non-conforming boundary elements, while domain integrals are evaluated within triangular cells. A test case is included to illustrate the described procedure.     

多跨连续梁自由振动分析的边界元法

核动力装置中的燃料元件棒和换热器传热管,常按多跨连续梁模型进行动力分析。本文用直接边界元法来分析这类多跨连续梁的特征值问题,推导出有轴向力作用和弹性支承条件的求解公式,并对压水堆燃料元件棒的固有频率和振型作了计算,得到较为满意的结果,可以看出,边界元法便于处理杆系结构问题,即使杆系的复杂度增加,而解题基本方法的复杂度并不增加。     

The why and how of finite elements

The ‘Why’ of the title reflects the desire in safety assessments for independent means of making realistic calculations for systems of complex shape. In principle the geometrical flexibility attainable by the deterministic finite element method being the equal of the stochastic Monte Carlo method.The development of the finite element method is traced to show how ideas from structural and fluid mechanics, the calculus of variations, functional analysis and the calculus of finite differences have been forged to provide a tool which minimizes the mismatch between the behaviour of a continuous system and that of a discrete model of the system assembled from finite elements. Geometrical flexibility of the model is achieved by the use of polygonal and curved elements. The behaviour of any point of an element is described in terms of its behaviour at discrete points or nodes of the element. In treating neutron transport the finite element method can be applied to phase-space, or as in this paper the spatial dependence can be treated by the use of finite elements in conjunction with expansions in orthogonal functions for the directional dependence.The mathematical formulation is based on a mixed parity form of the Boltzmann equation for one-group transport. The minimization of the mismatch between the system and its finite element model leads to a completely boundary-free maximum principle. This variational principle is also recast into a generalized least-squares principle. When the essential boundary conditions of the classical calculus of variations are imposed the well-known minimum and maximum principles for the even- and odd-parity second-order Boltzmann equations are obtained as special cases. The maximum principle for the second-order even-parity equation is used to demonstrate the precision and flexibility of the finite element method by solving the problems of a dog-legged duct in a shield and a cylindrical fuel element in a square lattice cell.The geometrical interpretation of the boundary-free maximum principle with the aid of a suitable Hilbert space then leads to completely boundary-free weighted residual or Galerkin schemes for both the first- and second-order forms of the Boltzmann equation. Imposing essential boundary conditions leads to classical schemes.The paper concludes with a sketch of finite element treatments of the multigroup Boltzmann equation.     

A Novel Boundary Element Method for Nonuniform Neutron Diffusion Problems

An advanced boundary element formulation has been proposed to solve the neutron diffusion equation (NDE) for a ‘nonuniform’ system. The continuous spatial distribution of a nuclear constant is assumed to be described using a polynomial function. Part of the constant term in the polynomial is left on the left-hand-side of the NDE, while the remainding is added to the fission source term on the right-hand-side to create a fictitious source. When the neutron flux is also expanded using a polynomial, the boundary integral equation corresponding to the NDE contains a domain integral related to the polynomial source. This domain integral is transformed into an infinite series of boundary integrals, by repeated application of the particular solution for a Poisson-type equation with the polynomial source. In two-dimensional, one-group test calculations for rectangular domains, the orthogonality of Legendre polynomials was used to determine the polynomial expansion coefficients. The results show good agreement with those obtained from finite difference computations in which the nonuniformity was approximated by a large number of material regions.     

Finite element nonlinear heat transfer analysis using a stable explicit method

The explicit stable method of Saulev is applied to nonlinear finite element heat conduction. Several nonlinear example problems are considered which include temperature-varying material properties and radiation boundary conditions.     

Anisotropic modelling of tube bundle flow during steam generator wet layup

Tube bundle flow can be considered as a porous medium flow and a fluid continuum can be established by introducing the porosity which is a ratio of fluid volume to total volume. Darcy's flow regime applies for the tube bundle flow of low Reynolds number during steam generator wet layup circulation. A general three-dimensional formulation appears as a steady-state heat conduction equation with source term and anisotropic conductivities. Solution to such an equation with appropriate boundary conditions can be obtained by any finite element computer program which solves anisotropic heat conduction problems. Capability of anisotropic modelling has been demonstrated by a sample problem of axisymmetric tube bundle flow with orthotropic hydraulic conductivities which are derived according to the existing empirical correlations for friction factors.     

Transient heat conduction problems in inhomogeneous media discretized by means of boundary-volume element

An integral equation formulation is presented for the transient heat conduction problems in inhomogeneous media. The material constants are assumed to be prescribed as arbitrary, continuous and differentiable functions of position vector. The governing integral equations are derived from the weighted residual statement of the problems in which the fundamental solution to the corresponding heat conduction problems in homogeneous media is used as the weight function. The whole domain of interest is discretized into a series of boundary-volume-time elements, and then a set of linear simultaneous equations are obtained. Their solutions yield the temperature in the whole domain as well as the heat flux on the boundary.     

Calculation of the heat flux in the lower divertor target plate using an infrared camera diagnostic system on the experimental advanced superconducting tokamak

During the discharging of Tokamak devices,interactions between the core plasma and plasma-facing components(PFCs) may cause exorbitant heat deposition in the latter. This poses a grave threat to the lifetimes of PFCs materials. An infrared(IR) diagnostic system consisting of an IR camera and an endoscope was installed on an Experimental Advanced Superconducting Tokamak(EAST) to monitor the surface temperature of the lower divertor target plate(LDTP) and to calculate the corresponding heat flux based on its surface temperature and physical structure, via the finite element method. First, the temperature obtained by the IR camera was calibrated against the temperature measured by the built-in thermocouple of EAST under baking conditions to determine the true temperature of the LDTP. Next, based on the finite element method, a target plate model was built and a discretization of the modeling domain was carried out. Then, a heat conduction equation and boundary conditions were determined. Finally, the heat flux was calculated. The new numerical tool provided results similar to those for DFLUX; this is important for future work on related physical processes and heat flux control.     

Three-Dimensional Analysis Method for Sloshing Behavior of FBR and Its Application to Uni- and Multi-Vessel Type FBRs

Abstract

A three-dimensional analysis method for sloshing behavior of FBR is developed. The method treats the coolant in a reactor vessel as a potential flow with moving liquid surfaces. The Laplace equation of velocity potential is solved by a boundary element method with its boundary conditions described by a Bernoulli equation.

The method is applied to analysis of sloshing behavior of uni- and multi-vessel type FBRs and results are presented.

The latter consists of vessels for the core, heat exchangers and pumps, all of which are connected by piping. In the uni-vessel type, heat exchangers and pumps are placed in the reactor vessel.     

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.     

Unsteady temperature distribution in bars of arbitrary cross section with heat generation

In general, numerical schemes such as the finite-difference method, the Monte-Carlo approach or the finite element technique must be applied when solving boundary value problems of heat conduction theory in the case of complex geometries quite common in nuclear reactor technology.It is shown in the present paper that an alternative analytical approach based on conformal mapping techniques and a variational formulation is quite convenient for the complicated domains considered herewith and for a type of unsteady state thermal field situation when heat generation takes place.     

Three-Dimensional Analysis Method for Sloshing Behavior of Fast Breeder Reactor and Its Application to Uni- and Multi-Vessel Type FBRs

A three-dimensional analysis method for sloshing behavior of fast breeder reactor (FBRs) is developed. The method treats the coolant in a reactor vessel as a potential flow with moving liquid surfaces. The Laplace equation of a velocity potential is solved by a boundary element method with its boundary condition described by a Bernoulli equation.

The vibration test results of a rectangular water pool are calculated by the method. Then, the method is applied to analysis of sloshing behavior of uni- and multi-vessel type FBRs. The latter consists of vessels for the core, heat exchangers and pumps. These vessels are connected by piping. In the case of the uni-vessel type FBR, heat exchangers and pumps are placed in the reactor vessel. The characteristics of sloshing behavior of both the reactors are presented.     

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.     

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.     

A solution of the neutron diffusion equation in hemispherical symmetry using the homotopy perturbation method

The homotopy perturbation method is used to formulate a new analytic solution of the neutron diffusion equation both for a sphere and a hemisphere of fissile material. Different boundary conditions are investigated; including zero flux on boundary, zero flux on extrapolated boundary, and radiation boundary condition. The interaction between two hemispheres with opposite flat faces is also presented. Numerical results are provided for one-speed fast neutrons in ²³⁵U. A comparison with Bessel function based solutions demonstrates that the homotopy perturbation method can exactly reproduce the results. The computational implementation of the analytic solutions was found to improve the numeric results when compared to finite element calculations.     

A hierarchical domain decomposition boundary element method with a higher order polynomial expansion for solving 2-D multiregion neutron diffusion equations

A hierarchical domain decomposition boundary element method (HDD-BEM) for solving the multiregion neutron diffusion equation (NDE) has been developed to reduce computation time. The boundary integral equations derived from NDEs defined in homogeneous subregions are discretized with higher order boundary elements. The neutron flux and the neutron currents on boundary elements are expanded by quadratic or cubic polynomials. This expansion allows a large decrease in the number of unknown variables compared with the conventional HDD-BEM with constant boundary elements and reduces the computation time greatly. To obtain high accuracy with a small number of unknowns it is important to assign suitable nodal points on the non-conforming boundary elements. Guidelines for the assignment of nodal points is presented through numerical analysis. The HDD-BEM with higher order boundary elements calculates at least 5 times faster than the conventional HDD-BEM with constant boundary elements and 30 times faster than the finite difference method. The improvements in computation time will enable an extension of the scope of application to a wider variety of problems in reactor analysis.