Transverse Integration Research Articles

A nodal integral method for quadrilateral elements

AbstractNodal integral methods (NIMs) have been developed and successfully used to numerically solve several problems in science and engineering. The fact that accurate solutions can be obtained on relatively coarse mesh sizes, makes NIMs a powerful numerical scheme to solve partial differential equations. However, transverse integration procedure, a step required in the NIMs, limits its applications to brick‐like cells, and thus hinders its application to complex geometries. To fully exploit the potential of this powerful approach, abovementioned limitation is relaxed in this work by first using algebraic transformation to map the arbitrarily shaped quadrilaterals, used to mesh the arbitrarily shaped domain, into rectangles. The governing equations are also transformed. The transformed equations are then solved using the standard NIM. The scheme is developed for the Poisson equation as well as for the time‐dependent convection–diffusion equation. The approach developed here is validated by solving several benchmark problems. Results show that the NIM coupled with an algebraic transformation retains the coarse mesh properties of the original NIM. Copyright © 2008 John Wiley & Sons, Ltd.

The analytic nodal diffusion solver ANDES in multigroups for 3D rectangular geometry: Development and performance analysis

In this work we address the development and implementation of the analytic coarse-mesh finite-difference (ACMFD) method in a nodal neutron diffusion solver called ANDES. The first version of the solver is implemented in any number of neutron energy groups, and in 3D Cartesian geometries; thus it mainly addresses PWR and BWR core simulations. The details about the generalization to multigroups and 3D, as well as the implementation of the method are given. The transverse integration procedure is the scheme chosen to extend the ACMFD formulation to multidimensional problems. The role of the transverse leakage treatment in the accuracy of the nodal solutions is analyzed in detail: the involved assumptions, the limitations of the method in terms of nodal width, the alternative approaches to implement the transverse leakage terms in nodal methods – implicit or explicit −, and the error assessment due to transverse integration. A new approach for solving the control rod “cusping” problem, based on the direct application of the ACMFD method, is also developed and implemented in ANDES. The solver architecture turns ANDES into an user-friendly, modular and easily linkable tool, as required to be integrated into common software platforms for multi-scale and multi-physics simulations. ANDES can be used either as a stand-alone nodal code or as a solver to accelerate the convergence of whole core pin-by-pin code systems. The verification and performance of the solver are demonstrated using both proof-of-principle test cases and well-referenced international benchmarks.

Analytic Function Expansion Nodal Method for Multigroup Diffusion Equations in Cylindrical (r,θ,z) Geometry

A coarse-mesh nodal method in cylindrical (r, θ,z) geometry, e.g., of pebble bed reactors, based on the analytic function expansion nodal (AFEN) methodology, is described in this paper. Two unique features are (a) no use of transverse integration—allowing a nodal scheme in (r, θ,z) geometry—and (b) nodal solution expressed in terms of analytic basis functions—leading to high accuracy and readily available reconstruction of homogeneous flux distributions. Additional features of multigroup formulation, two methods of void region treatment, and coarse-group-rebalance acceleration are implemented in the TOPS code and tested on several benchmark problems, including the Organisation for Economic Co-operation and Development/Nuclear Energy Agency PBMR-400 Benchmark Problem. The TOPS results are in excellent agreement with those of the VENTURE code, using significantly less computer time.

The analytic nodal method in cylindrical geometry

Nodal diffusion methods have been used extensively in nuclear reactor calculations, specifically for their performance advantage, but also for their superior accuracy. More specifically, the Analytic Nodal Method (ANM), utilising the transverse integration principle, has been applied to numerous reactor problems with much success. In this work, a nodal diffusion method is developed for cylindrical geometry. Application of this method to three-dimensional (3D) cylindrical geometry has never been satisfactorily addressed and we propose a solution which entails the use of conformal mapping. A set of 1D-equations with an adjusted, geometrically dependent, inhomogeneous source, is obtained. This work describes the development of the method and associated test code, as well as its application to realistic reactor problems. Numerical results are given for the PBMR-400 MW benchmark problem, as well as for a “cylindrisized” version of the well-known 3D LWR IAEA benchmark. Results highlight the improved accuracy and performance over finite-difference core solutions and investigate the applicability of nodal methods to 3D PBMR type problems. Results indicate that cylindrical nodal methods definitely have a place within PBMR applications, yielding performance advantage factors of 10 and 20 for 2D and 3D calculations, respectively, and advantage factors of the order of 1000 in the case of the LWR problem.

Influence of Orthogonal Overload on Human Vertebral Trabecular Bone Mechanical Properties

The aim of this study was to investigate the effects of overload in orthogonal directions on longitudinal and transverse mechanical integrity in human vertebral trabecular bone. Results suggest that the trabecular structure has properties that act to minimize the decrease of apparent toughness transverse to the primary loading direction. The maintenance of mechanical integrity and function of trabecular structure after overload remains largely unexplored. Whereas a number of studies have focused on addressing the question by testing the principal anatomical loading direction, the mechanical anisotropy has been overlooked. The aim of this study was to investigate the effects of overload in orthogonal directions on longitudinal and transverse mechanical integrity in human vertebral trabecular bone. T(12)/L(1) vertebral bodies from five cases and L(4)/L(5) vertebral bodies from seven cases were retrieved at autopsy. A cube of trabecular bone was cut from the centrum of each vertebral body and imaged by microCT. Cubes from each T(12)/L(1) and L(4)/L(5) pairs were assigned to either superoinferior (SI) or anteroposterior (AP) mechanical testing groups. All samples were mechanically tested to 10% apparent strain by uniaxial compression according to their SI or AP allocation. To elucidate the extent to which overload in orthogonal directions affects the mechanical integrity of the trabecular structure, samples were retested (after initial uniaxial compression) in their orthogonal direction. After mechanical testing in each direction, apparent ultimate failure stresses (UFS), apparent elastic moduli (E), and apparent toughness moduli (u) were computed. Significant differences in mechanical properties were found between SI and AP directions in both first and second overload tests. Mechanical anisotropy far exceeded differences resulting from overloading the structure in the orthogonal direction. No significant differences were found in mean UFS and mean u for the first or second overload tests. A significant decrease of 35% was identified in mean E for cubes overloaded in the SI direction and then overloaded in the AP direction. Observed differences in the mechanics of trabecular structure after overload suggests that the trabecular structure has properties that act to minimize loss of apparent toughness, perhaps through energy dissipating sacrificial structures transverse to the primary loading direction.

The Analytic Coarse-Mesh Finite Difference Method for Multigroup and Multidimensional Diffusion Calculations

In this work we develop and demonstrate the analytic coarse-mesh finite difference (ACMFD) method for multigroup—with any number of groups—and multidimensional diffusion calculations of eigenvalue and external source problems. The first step in this method is to reduce the coupled system of the G multigroup diffusion equations, inside any homogenized region (or node) of any size, to the G independent modal equations in the real or complex eigenspace of the G × G multigroup matrix. The mathematical and numerical analysis of this step is discussed for several reactor media and number of groups.As a second step, we discuss the analytical solutions in the general (complex) modal eigenspace for one-dimensional plane geometry, deriving the generalized Chao’s relation among the surface fluxes and the net currents, at a given interface, and the node-average fluxes, essential in the ACMFD method. We also introduce here the treatment of heterogeneous nodes, through modal interface flux discontinuity factors, and show the analytical and numerical application to core-reflector problems, for a single infinite reflector and for reflectors with two layers of different materials.Then, we address the general multidimensional case, with rectangular X-Y-Z geometry considered, showing the equivalency of the methods of transverse integration and incomplete expansion of the multidimensional fluxes, in the real or complex modal eigenspace of the multigroup matrix. A nonlinear iteration scheme is implemented to solve the multigroup multidimensional nodal problem, which has shown a fast and robust convergence in proof-of-principle numerical applications to realistic pressurized water reactor cores, with heterogeneous fuel assemblies and reflectors.

Two-dimensional nodal transport method for triangular geometry

The advanced nodal method for solving the multi-group neutron transport equation in two-dimensional triangular geometry is developed. To apply the transverse integration procedure, an arbitrary triangular node is transformed into a regular triangular node using coordinate transformation. The angular distributions of intra-node neutron fluxes and its transverse-leakage are represented by the S N quadrature set. The spatial distributions of neutron flux and source in the regular triangle are given approximately by an orthogonal quadratic polynomial, and the spatial expansion of transverse-leakage is approximated by a second-order polynomial. To establish a stable and efficient iterative scheme, the improved nodal-equivalent finite difference algorithm is used. The results for several benchmark problems demonstrate the higher capability of the method to yield the accurate results in significantly smaller computing times than those required by the standard finite difference method and the finite element spherical-harmonics method.

A nodal SN transport method for three-dimensional triangular- z geometry

A new nodal S N transport method has been developed to perform accurate transport calculation in three-dimensional triangular- z geometry, where arbitrary triangles are transformed into regular triangles via a coordinate transformation. The transverse integration procedure is applied to treat the neutron transport equation in the regular triangle. The neutron angular distributions of intra-node fluxes are represented using the S N quadrature set, and the spatial distributions of neutron fluxes and sources are approximated by a quadratic polynomial. The nodal-equivalent finite difference algorithm for 3D triangular geometry is applied to establish a stable and efficient iterative scheme. The present method was tested on four 3D Takeda benchmark problems published by the nuclear data agency (NEACRP), in which the first three problems are in XYZ geometry and the last one is in hexagonal- z geometry. The results of the present method agree well with those of the reference Monte-Carlo calculation method, the difference in k eff being less than 0.1%. This shows that multi-group reactor core/criticality problems can be accurately and effectively solved using the present method.

Intention‐driven modeling for flexible workflow applications

AbstractDuring the early 1990s, workflow technologies were the only ones to offer a transversal integration capacity to the enterprise applications, thus allowing the representation and the enactment of business processes. However, the formalisms developed for workflow specifications were almost systematically activity oriented. Consequently, the resulting process definitions have the advantage to be easily transformable in executable code, but the disadvantage of being prescriptive and rigid. Recent works underline the needs in term of flexible and adaptive workflows, whose execution can evolve according to situations that cannot always be prescribed. In fact, a flexible representation of business processes is an asset to maintain the fit between business processes and their supporting systems in an evolving environment. This article proposes a conceptual framework for an intention‐driven modeling of flexible workflow applications. The purpose of the underlying modeling formalism is to define an integration/orchestration for islands of business process chunks and their support systems in order to create and to maintain the fit between them. The modeling framework offers the ability to represent in the same business process definition the well‐structured process chunks as well as the ill‐structured or ad hoc ones. Copyright © 2005 John Wiley & Sons, Ltd.

The Real Corrections to the [formula omitted] impact factor: First Numerical Results

We have performed analytically the transverse momentum integrations in the real corrections to the longitudinal \gamma^*_L impact factor and carried out numerically the remaining integrations. I outline the analytical integration and present the numerical results: we have performed a numerical test and computed those parts of the impact factor that depend upon the energy scale s_0.

Modified Nodal Integral Method for the Three-Dimensional, Time-Dependent, Incompressible Navier-Stokes Equations

A modified nodal integral method (MNIM) for two-dimensional, time-dependent Navier-Stokes equations is extended to three dimensions. The nodal integral method is based on local transverse integrations over finite size cells that reduce each partial differential equation to a set of ordinary differential equations (ODEs). Solutions of these ODEs in each cell for the transverse-averaged dependent variables are then utilized to develop the difference schemes. The discrete variables are scalar velocities and pressure, averaged over the faces of bricklike cells. The development of the MNIM is different from the conventional nodal method in two ways: (a) it is Poisson-type pressure equation based and (b) the convection terms are retained on the left side of the transverse-integrated equations and thus contribute to the homogeneous part of the solution. The first feature leads to a set of symmetric transverse-integrated equations for all the velocities, and the second feature yields distributions of constant + linear + exponential form for the transverse-averaged velocities. The scheme is tested on three-dimensional lid-driven cavity problems in cube- and prism-shaped cavities. Results obtained using the MNIM on fairly coarse meshes are comparable with reference solutions obtained using much finer meshes.

Next-to-leading-order corrections to theγ*impact factor: First numerical results for the real corrections toγL*

We analytically perform the transverse momentum integrations in the real corrections to the longitudinal \gamma*_L impact factor. The resulting integrals are Feynman parameter integrals, and we provide a MATHEMATICA file which contains the integrands. The remaining integrals are carried out numerically. We perform a numerical test, and we compute those parts of the impact factor which depend upon the energy scale s_0: they are found to be negative and, with decreasing values of s_0, their absolute value increases.

High-Order Analytical Nodal Method for the Multigroup Diffusion Equations

We develop a high-order analytical nodal method for the multigroup diffusion equations. Based on the transverse integration procedure, the discrete 1D equations are analytically approximated using the combined direct algebraic evaluation of trigonometric functions of multigroup matrices, and the truncated Legendre series. The remaining Legendre coefficients of the transverse leakage moments are determined exactly in terms of the different neutron flux moments order. The self-consistent is guaranteed. In the weighted balance equations, the transverse leakage moments are linearly written in terms of the partial currents, facial and centered fluxes moments. Furthermore, as the order increases, the neutron balance in each node and the coupling between the adjacent cell are reinforced. The efficacy of the method is shown for 2D-PWR and 2D-LMFBR benchmark problems.

ICONE11-36501 ADVANCES IN THE SOLUTION OF THREE-DIMENSIONAL NODAL NEUTRON TRANSPORT EQUATION

In this paper we study the three-dimensional nodal discrete-ordinates approximations of neutron transport equation in a convex domain with piecewise smooth boundaries. We use the combined collocation method of the angular variables and nodal approach for the spatial variables^1. By nodal approach we mean the iterated transverse integration of the S_N equations. This procedure leads to the set of one-dimensional averages angular fluxes in each spatial variable. The resulting system of equations is solved with the LTS_N method, first applying the Laplace transform to the set of the nodal S_N equations and then obtaining the solution by symbolic computation. We include the LTS_N method by diagonalization to solve the nodal neutron transport equation and then we outline the convergence of these nodal-LTS_N approximations with the help of a norm associated to the quadrature formula used to approximate the integral term of the neutron transport equation^<(2,3)> We give numerical results obtained with an algebraic computer system (for N up to 8) and with a code for higher values of N. We compare our results for the geometry of a box with a source in a vertex and a leakage zone in the opposite vertex with others techniques used in this problem. code for higher values of N. We compare our results for the geometry of a box with a source in a vertex and a leakage zone in the opposite vertex with others techniques used in this problem.

Building man–man–machine synergies: Experiences from the Vanderbilt and Geneva clinical information systems

Objectives: To demonstrate how a transition to an open hospital information system architecture can help foster stronger collaborations between computer-based clinical tools and the network of care process stakeholders. Methods: Description of two evolution strategies towards an open, component-based architecture: the vertical decomposition of a monolithic system, and the transversal integration of foundation components such as terminology servers, documentation management, prescription, scheduling, workflow and notification engines. Conclusion: The progressive migration of production clinical systems towards a component-based architecture is feasible at a reasonable cost and leads to substantial benefits in terms of user acceptance through participative design and collaborative maintenance of knowledge bases, as well as improved ability to evolve, scale and share individual components. A taxonomy of the components of a health information system and their interactions should be developed to realize the collaborative potential of a marketplace of interoperable components.

Higher order analytical nodal methods in response-matrix formulation for the multigroup neutron diffusion equations

A higher analytical nodal method for the multigroup neutron diffusion equations, based on the transverse integration procedure, is presented. The discrete 1D equations are cast with the interface partial current techniques in response matrix formalism. The remaining Legendre coefficients of the transverse leakage moment are determined exactly in terms of the different neutron flux moments order in the reference node. In the weighted balance equations, the transverse leakage moments are linearly written in terms of the partial currents, facial and centered fluxes moments. The self-consistent is guaranteed. Furthermore, as the order k increase the neutronic balance in each node and the copulate between the adjacent cell are reinforced. The convergence order in L 2-norm is of O( h k+3− δ k0 ) under smooth assumptions. The efficacy of the method is showed for 2D-PWR, 2D-IAEA LWR and 2D-LMFBR benchmark problems.

Off-zone-center or indirect band-gap-like hole transport in heterostructures

Unintuitive hole transport phenomena through heterostructures are presented. It is shown that for large bias ranges the majority of carriers travel outside the $\ensuremath{\Gamma}$ zone center (i.e., more carriers travel through the structure at an angle than straight through). Strong interaction of heavy-, light-, and split-off hole bands due to heterostructure interfaces present in devices such as resonant tunneling diodes, quantum-well photodetectors, and lasers are shown to be the cause. The result is obtained by careful numerical analysis of the hole transport as a function of the transverse momentum k in a resonant tunneling diode within the framework of a $\mathrm{sp}3s*$ second-nearest-neighbor tight-binding model. Three independent mechanisms that generate off-zone-center current flow are explained: (1) nonmonotonic (electronlike) hole dispersion, (2) lighter quantum well than emitter effective masses, and (3) strongly momentum-dependent quantum-well coupling strength due to state anticrossings. Finally a simulation is compared to experimental data to exemplify the importance of a full numerical transverse momentum integration versus a Tsu-Esaki approximation.

Triangle anomaly in triple-Regge limits

Reggeized gluon interactions due to a single quark loop are studied in the full triple-regge limit and in closely related helicity-flip helicity-pole limits. Triangle diagram reggeon interactions are generated that include local axial-vector effective vertices. It is shown that the massless quark triangle anomaly is present as a chirality-violating infra-red divergence in the interactions generated by maximally non-planar Feynman diagrams. An asymptotic dispersion relation formalism is developed which provides a systematic counting of anomaly contributions. The asymptotic amplitude is written as a sum over dispersion integrals of triple discontinuities, one set of which is unphysical and can produce chirality transitions. The physical-region anomaly appears in the generalized real parts, determined by multi-regge theory, of the unphysical discontinuities. The amplitudes satisfy a signature conservation rule that implies color parity is not conserved by vertices containing the anomaly. In the scattering of elementary quarks or gluons the signature and color parity of the exchanged reggeon states are such that the anomaly cancels. At lowest-order, it cancels in individual diagrams after the transverse momentum integrations are performed.

A Two-Dimensional Intranodal Flux Expansion Method for Hexagonal Geometry

A new nodal method HEXNEM2 for hexagonal geometry is described. The method is based on a two-dimensional expansion of the intranodal fluxes. Polynomials up to the second order and exponential functions are used in each group. By this method, the singular terms occurring in the transverse integration methods are avoided. Side-averaged and corner-point values of fluxes and currents are used for the coupling of nodes. A calculation scheme for the outgoing partial currents at the sides and similar expressions for the corners from given incoming values are used in the inner iteration, which gives a fast-running scheme. The method is tested against two-dimensional hexagonal benchmark problems for the VVER-type reactors. The results show that the multiplication factor and nodal powers are predicted accurately. A considerable improvement can be shown in the results for the VVER-1000 benchmarks compared with the method developed previously for the code DYN3D and the simpler method HEXNEM1.

Mesh‐centered finite differences from nodal finite elements for elliptic problems

After it is shown that the classical five-point mesh-centered finite difference scheme can be derived from a low-order nodal finite element scheme by using nonstandard quadrature formulae, higher-order block mesh-centered finite difference schemes for second-order elliptic problems are derived from higher-order nodal finite elements with nonstandard quadrature formulae as before, combined to a procedure known as “transverse integration.” Numerical experiments with uniform and nonuniform meshes and different types of boundary conditions confirm the theoretical predictions, in discrete as well as continuous norms. © 1998 John Wiley & Sons, Inc. Numer Methods Partial Differential Eq 14: 439–465, 1998