SPN Equations Research Articles

Development of a diffusive simplified double P[formula omitted] approximation to the neutron transport equation

The neutron transport equation models the distribution of the neutron power inside a reactor core. The complexity of integrating this equation implies the use of some angular approximations. Discrete ordinates and spherical harmonics equations (PN equations) are commonly used to this purpose, but even when these approaches are used, the resulting set of equations requires high computational demands. An alternative is to develop simplified formulations of them, such as the simplified spherical harmonics equations (SPN equations), which provide accurate results. The spherical harmonic equations are based on the regularity of solution in terms of the angular dependence. However, in some reactor configurations, the angular fluxes can be discontinuous. For that, the double spherical harmonics equations, where angular fluxes are expanded in terms of two sets of Legendre polynomials, were studied. In this work, a simplified treatment of them is developed that leads to a diffusive set of equations, the simplified double spherical harmonics equation (SDPN equations). Numerical results show that the SDPN approximation provides faster and more accurate results than the SPN equations for some problems with strong variations of the angular neutron flux.

Computing enhancement of the nonlinear SPN approximations of radiative heat transfer in participating material

Anisotropic mesh adaptation is an efficient procedure for controlling the output error of finite element simulations, particularly when used for three-dimensional problems. In this paper, we present an enhanced computational algorithm based on an anisotropic mesh adaptation for nonlinear SPN approximations of radiative heat transfer in both two- and three-dimensional enclosures. Using an asymptotic analysis for the optical scale in the radiative transfer, the integro-differential equation is replaced by a series of partial differential equations of elliptic type. The nonlinear coupling between the heat transfer and radiation in the SPN equations does not depend on the direction ordinates. In an anisotropic participating media, internal boundary layers with different magnitudes occur for each direction and developing an efficient numerical algorithm to accurately resolve them is a challenging task. In the present study, we propose an adaptive finite element method using an efficient hierarchical error estimator. A second-order scheme is used for the time integration and a Newton-type solver is implemented for the fully coupled nonlinear system. The proposed method has the potential to use large timesteps in the simulations for radiative heat transfer in anisotropic media at high temperatures. To demonstrate the viability of the method, several examples are presented for two- and three-dimensional problems. The numerical results confirm the capability of the proposed method to efficiently solve the nonlinear SPN approximations of radiative transfer in anisotropic media.

Time-dependent simplified spherical harmonics formulations for a nuclear reactor system

The steady-state simplified spherical harmonics equations (SPN equations) are a higher order approximation to the neutron transport equations than the neutron diffusion equation that also have reasonable computational demands. This work extends these results for the analysis of transients by comparing of two formulations of time-dependent SPN equations considering different treatments for the time derivatives of the field moments. The first is the full system of equations and the second is a diffusive approximation of these equations that neglects the time derivatives of the odd moments. The spatial discretization of these methodologies is made by using a high order finite element method. For the time discretization, a semi-implicit Euler method is used. Numerical results show that the diffusive formulation for the time-dependent simplified spherical harmonics equations does not present a relevant loss of accuracy while being more computationally efficient than the full system.

A block Arnoldi method for the SPN equations

ABSTRACTThe simplified spherical harmonics equations are a useful approximation to the stationary neutron transport equation. The eigenvalue problem associated with them is a challenging problem from the computational point of view. In this work, we take advantage of the block structure of the involved matrices to propose the block inverse-free preconditioned Arnoldi method as an efficient method to solve this eigenvalue problem. For the spatial discretization, a continuous Galerkin finite element method implemented with a matrix-free technique is used to keep reasonable memory demands. A multilevel initialization using linear shape functions in the finite element method is proposed to improve the method convergence. This initialization only takes a small percentage of the total computational time. The proposed eigenvalue solver is compared to the standard power iteration method, the Krylov-Schur method and the generalized Davidson method. The numerical results show that it reduces the computational time to solve the eigenvalue problem.

A new and rigorous SPN theory – Part II: Generalization to GSPN

A new SPN theory valid for piecewise homogeneous regions was rigorously derived in a paper published last year by the author. The new theory does not need the assumption of locally 1D behavior near a surface. The resulting interface and the boundary conditions, different from the ad hoc conditions conventionally assumed, contain higher order derivatives of the solution functions to the SPN equations.This new SPN theory was presented in the previous paper for the mono-energetic case with generic isotropic neutron sources in piecewise homogeneous regions. In this sequel paper the theory is generalized to the generic case of multi-energy groups including anisotropic neutron sources as well. Although the multi-group extension is straightforward, the inclusion of anisotropic sources poses subtle problems. To be valid for any arbitrary linear anisotropic external source a Generalized SP3 (GSP3) theory is proposed. GSP3 clarifies how to decompose a linearly anisotropic source via Helmholtz decomposition so that its irrotational component can be properly incorporated into SP3. GSP3 is more accurate than SP3. It includes two additional diffusion type equations decoupled from the two original SP3 equations, with in total eight instead of four interface or boundary conditions. The GSP3 equations are all of diffusion type and the additional interface and boundary conditions are manageable for practical engineering applications.By generalizing the Helmholtz decomposition to higher order tensors and using a powerful lemma embedded in a Boris Davison’s 1958 paper, we can formulate a comprehensive theory for the generic GSPN, which can accommodate any spatially differentiable anisotropic neutron sources of any order in anisotropy. The only restriction to GSPN is piecewise homogeneity. GSPN theory provides various levels of approximations to PN, from the simplest SPN all the way to the rigorous equivalency to PN. GSPN contains decomposed independent sets of diffusion type second order differential equations without any crossed variable derivatives, which are very easy to solve. But the interface and boundary conditions are more complicated where the solutions to the independent sets of equations do get coupled. An interesting byproduct of GSPN is that it provides the recipe of how to decompose and incorporate arbitrary anisotropic neutron sources into the SPN equations via the use of the generalized Helmholtz decomposition. The general formulation for the GSPN theory is presented, although specific details are given only for GSP3.

Mathematical and Numerical Validation of the Simplified Spherical Harmonics Approach for Time-Dependent Anisotropic-Scattering Transport Problems in Homogeneous Media

ABSTRACTIn this work, we demonstrate theoretically and numerically the equivalence between the SPN and PN approximation for time-dependent problems in homogeneous media with anisotropic scattering. Our theory extends the solid harmonics derivation, which was used by Ackroyd et al. to derive the steady-state SPN equations, to transient problems. The derivation expands the angular flux in ordinary surface harmonics but uses harmonic polynomials to generate additional surface spherical harmonic terms to be used in Galerkin projection. Also, we use the line source problem and McClarren's “box” problem to demonstrate such equivalence numerically. Both problems were initially proposed for isotropic scattering, but here we add higher order scattering moments to them. Numerical results show that the difference between the SPN and PN scalar flux solution is at the roundoff level.

A new and rigorous SPN theory for piecewise homogeneous regions

A new SPN theory valid for piecewise homogeneous regions is rigorously derived without mathematical approximations. The resulting SPN equations are the same as the conventional ones, but both the interface and the boundary conditions are different from the ad hoc conditions conventionally assumed. The new theory does not need the assumption of locally 1D behavior near a surface.Gelbard pointed out in 1960 that SPN is rigorously valid for an infinite domain 3D problem with constant total cross-section and arbitrarily distributed isotropic sources, and speculated that SPN could be a good approximation for practical applications. The Gelbard’s example does not involve any interface or boundary condition. For a generic problem with piecewise homogeneous regions, even though the SPN equations can still be rigorously derived for each homogeneous region, the interface or boundary condition poses the problem for generalizing the rigorous SPN theory. In this paper we point out that in each of the homogeneous regions, the solution to the SPN equations can be used to rigorously reconstruct an angular flux distribution which is a particular solution to the PN equations. The complete PN angular flux solution is the sum of this particular SPN angular flux solution and a set of additional functions that are solution to the homogeneous part of the PN equations. This particular SPN angular flux solution alone does not have enough independent functions to fulfill all the PN interface or boundary conditions. In this case, one can only satisfy a subset of the conditions. Because this subset of conditions cannot be decoupled from the full set of conditions, the whole core SPN solution so obtained cannot provide exactly the same scalar flux solution as that given by the PN equations. Nevertheless if this particular SPN angular flux solution is to be used alone, its corresponding interface and boundary conditions can be explicitly derived without introducing mathematical approximations. The resulting conditions are different from the conventionally used ad hoc conditions for SPN. The new interface and boundary conditions involve additional terms of higher order derivatives of the SPN solution and are consistent with what Selengut found for SP3 in 1970. The arguments and techniques needed to establish the above theory can in fact be all deduced from the detailed PN theory formulation for piecewise homogeneous regions that was worked out by Davison in 1957 and 1958.This new SPN theory is valid for the multi-group case as well and can be generalized to the case of anisotropic scattering. It may be a good candidate for practical applications to 3D pin-cell meshed whole core calculations, if the pin-cell meshes can be properly homogenized. The new interface and boundary conditions can be implemented in conventional SPN codes but require more work to revise the coupling equations, resulting in new SPN nodal equations. The discontinuity factors needed for using homogenized meshes can be generated without ambiguity by using the particular SPN angular flux solution as well. A physical interpretation for the SPN theory is suggested. It is believed that the missing steps are now filled to have a sound and complete SPN theory.

Analytical solution of the simplified spherical harmonics equations in spherical turbid media

We present for the first time an analytical solution for the simplified spherical harmonics equations (so-called SPN equations) in the case of a steady-state isotropic point source inside a spherical homogeneous absorbing and scattering medium. The SPN equations provide a reliable approximation to the radiative transfer equation for describing light transport inside turbid media. The SPN equations consist of a set of coupled partial differential equations and the eigen method is used to obtain a set of decoupled equations, each resembling the heat equation in the Laplace domain. The equations are solved for the realistic partial reflection boundary conditions accounting for the difference in refractive indices between the turbid medium and its environment (air) as occurs in practical cases of interest in biomedical optics. Specifically, we provide the complete solution methodology for the SP3, which is readily applicable to higher orders as well, and also give results for the SP5. This computationally easy to obtain solution is investigated for different optical properties of the turbid medium. For validation, the solution is also compared to the analytical solution of the diffusion equation and to gold standard Monte Carlo simulation results. The SP3 and SP5 analytical solutions prove to be in good agreement with the Monte Carlo results. This work provides an additional tool for validating numerical solutions of the SPN equations for curved geometries.

A new SPn theory formulation with self-consistent physical assumptions on angular flux

The conventional SPn theory cannot provide the explicit angular flux solution. Chao and Yamamoto (2012) proposed the explicit angular flux representation for the SPn theory as cylindrically symmetric with respect to the net current direction at any point in space, such that the angular flux is an expansion in Legendre polynomials of the cosine of the polar angle with respect to the local net current. Such a model however cannot lead to the SPn equations without further ad hoc assumptions because the multiple directions of the spatial gradients of the flux moments cannot be all parallel to the net current. In this paper we relax the assumption of the total angular flux being locally one-dimensional and generalize it to each of the flux moments being locally one-dimensional along the direction of the spatial gradient of each individual flux moment. The angular distribution of the nth order flux moment is the nth order Legendre polynomial of the cosine of the polar angle with respect to the direction of the spatial gradient of the nth order flux moment, ∇ϕn(r). With this physical model one can rigorously derive the equations for the current and the boundary conditions. The SPn equations can also be derived with the additional assumption of the total cross-section being locally constant, which is practically always valid when the spatial variation is discretized in numerical calculations. However the boundary conditions turn out to be different from the conventional ones, containing some non-linear factors. The internal interface boundary conditions are not affected by the non-linear factors as they cancel out on the interface. But the external boundary condition does get affected by the non-linear factors. The effect of non-linear factors is of higher order and if neglected, the external boundary condition also reduces to the conventional one. The non-linear external boundary condition can be iteratively updated to estimate the correction effect. A numerical calculation problem is suggested to test the new SPn theory and to assess the non-linear effect.

Relationship of the SPN_AN method to the even-parity transport equation

There are two distinct aspects to this paper. Firstly, it is shown that the AN form of the SPN equations can be derived from a suitably discretised form of the even parity transport equation. The procedure for doing this is explained and some limitations of SPN_AN theory are illustrated. Secondly, to show the magnitude of the errors in AN theory we have taken the problem of an infinitely repeating lattice in which the cell structure is symmetric but can contain any number of sub-regions. The regions can be elliptical and/or rectangular. Because of the repeating lattice, we can express the flux in the cell in the form of a Fourier series. The expansion coefficients of the series may then be obtained from a set of linear equations. It is found that the series converges very slowly and a large number of terms is required which can significantly increase the computational time required for accurate solutions. Use is made of a high order SN (N=200) method to compare and test convergence of the Fourier method for an exact case and for the SPN equivalent.

Efficient solution of the simplified [formula omitted] equations

In this paper we show new solver strategies for the multigroup SPN equations for nuclear reactor analysis. By forming the complete matrix over space, moments, and energy, a robust set of solution strategies may be applied. Power iteration, shifted power iteration, Rayleigh quotient iteration, Arnoldi's method, and a generalized Davidson method, each using algebraic and physics-based multigrid preconditioners, have been compared on the C5G7 MOX test problem as well as an operational pressurized water reactor model. Our results show that the most efficient approach is the generalized Davidson method, which is 30–40 times faster than traditional power iteration and 6–10 times faster than Arnoldi's method.

Elliptic formulation of the Simplified Spherical Harmonics Method in radiative heat transfer

The SimplifiedPN-Approximation was introduced some 50years ago, as an intuitive three-dimensional extension to the one-dimensional slab PN-formulation. The set of simplified-PN or SPN equations was formulated, such that they reduced to the standard PN-approximation for a one-dimensional slab, resulting in 2(N+1) simultaneous first order differential equations, subject to a modified set of Marshak boundary conditions. In this article the method is developed further for radiative heat transfer applications by reducing the governing equations to a small set of simultaneous second-order, elliptic PDEs, similar to the simple P1 equation. Each of the equations is subject to P1-type boundary conditions. Several two-dimensional examples are presented, including scattering, strongly inhomogeneous temperatures and absorption coefficients.

Iterative performance of various formulations of the SPN equations

In this paper, the Standard, Composite, and Canonical forms of the Simplified P"N (SP"N) equations are reviewed and their corresponding iterative properties are compared. The Gauss-Seidel (FLIP), E...

Domain Decomposition for the SPN Solver MINOS

In this article we present a domain decomposition method for the mixed SPN equations, discretized with Raviart-Thomas-Nédélec finite elements. This domain decomposition is based on the iterative Schwarz algorithm with Robin interface conditions to handle communications. After having described this method, we give details on how to optimize the convergence. Finally, we give some numerical results computed in a realistic 3D domain. The computations are done with the MINOS solver of the APOLLO3® code.

Diffuse optical tomographic imaging of biological media by time-dependent parabolic SPN equations: a two-dimensional study

We investigate the problem of retrieving the optical properties (absorption and scattering) of biological tissue from a set of optical measurements. A diffuse optical tomography (DOT) algorithm that incorporates constrained optimization methods is implemented. To improve image quality, the DOT algorithm exploits full time-domain data. The time-dependent parabolic simplified spherical harmonics equations (TD-pSPN) are used as the forward model. Time-dependent adjoint variables are resorted to in the calculation of the gradient of the objective function. Several numerical experiments for small geometric media with embedded inclusions that mimic small animal imaging are performed. In the experiments, optical coefficient values are varied in the range of realistic values for the near-infrared spectrum, including high absorption values. Single and multiparameter reconstructions are performed with the diffusion equation and higher orders of the TD-pSPN equations. The results suggest the DOT algorithm based on the TD-pSPN model outperforms the DE, and accurately reconstructs optical parameter distributions of biological media both spatially and quantitatively.

The Simplified Spherical Harmonics Method For Radiative Heat Transfer

The Simplified PN-Approximation was introduced some 50 years ago, as an intuitive three-dimensional extension to the one-dimensional slab PN-formulation. The set of simplified-PN or SPN equations was formulated, such that they reduced to the standard PN-approximation for a one-dimensional slab, resulting in 2(N+1) simultaneous first order differential equations, subject to a modified set of Marshak boundary conditions. In this article the method is developed further for radiative heat transfer applications by reducing the governing equations to a small set of simultaneous second-order, elliptic PDEs, similar to the simple P1 equation. Each of the equations is subject to P1-type boundary conditions. Several two-dimensional examples are presented, including scattering, strongly inhomogeneous temperatures and absorption coefficients.

Goal-Oriented h-Adaptivity for the Multigroup SPN Equations

We investigate application of goal-oriented mesh adaptivity to the SPN multigroup equations. This technique utilizes knowledge of the computational goal and combines it with mesh adaptivity to accurately and rapidly compute quantities of interest. Specifically, the local error is weighted by the importance of a given cell toward the computational goal, resulting in appropriate goal-oriented error estimates. Even though this approach requires the solution of an adjoint (dual) problem, driven by a specific source term for a given quantity of interest, the work reported here clearly shows the benefits of such a method.We demonstrate the level of accuracy this method can achieve using two-dimensional and three-dimensional numerical test cases for one-group and two-group models and compare results with more traditional mesh refinement and uniformly refined meshes. The test cases consider situations in which the radiative flux of a source is shielded and are designed to prototypically explore the range of conditions under which our methods improve on other refinement algorithms. In particular, they model strong contrasts in material properties, a situation ubiquitous in nuclear engineering.

Grenade - A coarse-mesh reactor physics program to solve the static diffusion equation for neutrons

The paper deals with general properties of the odd-order SPN equations. The equivalence between these equations and the so-called AN equations is demonstrated. The AN equations have the simple structure of a system of multigroup diffusion equations and are easily transformed into a set of boundary integral equations, which allows to apply a boundary element-response matrix solution technique and obtain a proper setting for the proofs of the boundary and interface conditions which are to be used. Some sharper results for the special class of the transport problems in which the total cross section is constant are discussed.