Nodal Method Research Articles

Application of stiffness confinement method within variational nodal method for solving time-dependent neutron transport equation

The stiffness confinement method (SCM) is implemented in the variational nodal method (VNM) framework to solve time-dependent multi-group neutron transport equations. With frequency-transform, the time-dependent neutron equation is decoupled from the precursor equation and transformed into a dynamic eigenvalue problem which can be solved with the in-house VNM code VITAS. An improved time-source iteration method within the VNM-SCM framework is proposed to accelerate constructing response matrices and reduces memory usage. Based on the VNM-SCM scheme, the kinetics module of VITAS is developed for the Cartesian and hexagonal-z geometry and verified by several benchmark problems.Numerical results demonstrate that VNM-SCM can perform accurate and efficient transient calculations with large time-step sizes. For the two-dimensional TWIGL benchmark, the VNM-SCM produces a maximum core power error of -0.57% with the time-step size of 20 ms. For the three-dimensional LMW benchmark, the VNM-SCM results in a maximum core power error of 0.95% with the time-step size of 2 s. The VNM-SCM also yields an accurate neutron flux prediction for the hexagonal-z geometry problem. In addition, the effect of the angular expansion is investigated systematically. Examinations with the three-dimensional LMW benchmark display an error reduction in the core power error from 0.131% to 0.011% from P1 to P3 interfacial angular expansion, with a substantial computing time increase of 6.14 hours. The study indicates that the high-order angular expansion significantly increases the computing time but contributes little to the accuracy of the core power. Besides, it was found that the initial steady-state calculation dominates the shape error of the flux in the transient calculation.

Improved macroscopic depletion in 2-D nodal analysis by 2x2 albedo-corrected parameterized equivalence constants method

This work introduces a 2x2 albedo-corrected parameterized equivalence constants (APEC) method for the macroscopic depletion of nuclear reactors in 2-D nodal analysis to overcome the inherent limitations of the conventional two-step method. A new 2x2 APEC leakage correction that improves fuel assembly (FA) homogenized group constants (HGCs) with a 2x2 FA subdivision is proposed as an improvement on the previous APEC method correcting the HGCs with a 1x1 FA subdivision. Both burnup-independent and -dependent APEC functions are devised by performing various color-set depletion calculations using a transport lattice code. The APEC method is implemented to an in-house macroscopic depletion nodal code based on the nodal expansion method (NEM). For macroscopic depletion, Xe-135 and Sm-149 transient models are proposed based on simplified Xe-135 and Sm-149 decay chains. In addition to the lattice burnup, power-density is newly introduced as an additional independent parameter in the lattice calculations to consider a history effect in HGCs. Both UOX-loaded and partially MOX-loaded SMR benchmarks are analyzed to assess the performance of the new APEC method. Furthermore, the general applicability of the APEC correction scheme is also evaluated using random variants of the two reference cores, including large-sized PWR problems. Results show that the new burnup-dependent APEC leakage correction can substantially improve the nodal equivalence in terms of the effective multiplication factor and fuel assembly-wise power distribution.

Parallel Higher Order DGTD and FETD for Transient Electromagnetic-Circuital-Thermal Co-Simulation

A hybrid higher order discontinuous Galerkin time-domain (DGTD) method and finite-element time-domain (FETD) method with parallel technique is proposed for electromagnetic (EM)–circuital–thermal co-simulation in this article. For electromagnetic simulation, DGTD method with higher order hierarchical vector basis functions is used to solve Maxwell equation. Circuit simulation is carried out by modified nodal analysis method. For thermal simulation, FETD method with higher order interpolation scalar basis functions is adopted to solve heat conduction equation. To implement electromagnetic–circuital–thermal co-simulation, the electromagnetic and circuital equations are strongly coupled through voltages, currents, and electric fields at the lumped ports first. Then the electromagnetic and thermal equations are weakly coupled with electromagnetic loss and temperature-dependent medium parameters. Finally, large-scale parallel technique is used to accelerate the process of multiphysics simulation. Numerical results are given to validate the correctness and capability of the proposed electromagnetic–circuital–thermal co-simulation method.

A mathematical model for the corneal transparency problem

AbstractUnderstanding the physical basis of corneal transparency has been a subject of interest amongst physicists, basic scientists and ophthalmologists. Impairment of corneal clarity is a significant cause of visual morbidity worldwide. Thus, it is essential to understand the mechanisms behind corneal transparency and how the alterations due to corneal pathologies affect vision. We use Maxwell’s equations to model light propagation in ocular tissues and a nodal discontinuous Galerkin method combined with an explicit Runge-Kutta method to simulate light propagation in normal and pathological corneas. Our simulation results illustrate that an increase in the diameter of some fibres causes an increase in backscattering. Thus, these may represent some of the physical changes in the cornea that might result in loss of transparency and visual morbidity.

A new coupling method based on MOC and the SN nodal method for the neutron transport calculations

The ray spacing must be fine enough to traverse every region of the reactor to get accurate results when using the method of characteristics. However, when calculating some research reactors, the active zone is relatively small compared to the entire problem. It would be time-consuming and memory costing to apply MOC for the entire geometry. A new approach that couples MOC and the SN nodal method is proposed and implemented to solve those issues. The calculation region is divided into the MOC domain, which includes active zone or complex structures, and the SN domain, such as the moderator and reflector region. The MOC and SN nodal methods are coupled by the interface angular flux. The 2D C5G7 benchmark and a 2D JRR-3 plate-type fuels reactor problem are tested to demonstrate the accuracy and efficiency of the coupling method, and the numerical results show that the coupling method achieves superior efficiency.

Diagnostic accuracy of de-escalated surgical procedure in axilla for node-positive breast cancer patients treated with neoadjuvant systemic therapy: A systematic review and meta-analysis.

More initial clinical node-positive breast cancer patients achieve axillary pathological complete response (ax-pCR) after neoadjuvant systemic therapy (NST). Restaging axillary status and performing de-escalated surgical procedures to replace routine axillary lymph nodes dissection (ALND) is urgently needed. Targeted axillary lymph node biopsy (TLNB) is a novel de-escalated surgical strategy marking metastatic axillary nodes before NST and targeted dissection and biopsy intraoperatively to tailor individual axillary management. This study provided a systematic review and meta-analysis to evaluate the feasibility and diagnosis accuracy of TLNB. Prospective and retrospective clinical trials on TLNB were searched from Pubmed, Embase, and Cochrane. Identification rate (IFR), false-negative rate (FNR), negative predictive value (NPV), and rate of ax-pCR were the outcomes of this meta-analysis. One thousand nine hundred and twenty patients attempted TLNB, with an overall IFR of 93.5% (95% confidence interval [CI] 90.1%-96.2%). IFR of three nodal marking methods, namely iodine seeds, clips, and carbon dye, was 95.6% (95% CI 91.2%-98.7%), 91.7% (95% CI 87.3%-95.4%), and 97.1% (95% CI 89.1%-100.0%), respectively. Of them, 847 patients received ALND, with an overall FNR of 5.5% (95% CI 3.3%-8.0%), and NPV ranged from 90.1% to 96.1%. Regression analysis showed that the overlap of targeted and sentinel biopsied nodes might associate with IFRs and FNRs. TLNB is a novel, less invasive surgical approach to distinguish initial node-positive breast cancer that achieves negative axillary conversion after NST. It yields an excellent IFR with a low FNR and a high NPV. A combination of preoperative imaging, intraoperative TLNB with SLNB, and postoperative nodal radiotherapy might affect the future treatment paradigm of primary breast cancer with nodal metastases.

2D Fourier finite element formulation for magnetostatics in curvilinear coordinates with a symmetry direction

We present a numerical method for the solution of three-dimensional linear magnetostatic problems by embedding their geometry into a symmetric domain and Fourier expansion in the symmetry coordinate, leading to a set of decoupled two-dimensional problems. Special cases include axial and translational symmetry. To keep the treatment universal, a framework for the Fourier decomposition of the vector potential formulation is developed in the covariant formalism with general curvilinear coordinates. Under the restriction of homogeneous material parameters along the symmetry direction, but not on fields and current densities, this leads to a decoupled set of two-dimensional equations in both, zeroth (non-oscillatory) and non-zero (oscillatory) harmonics. For the latter it is possible to eliminate one component of the vector potential resulting in a fully transverse vector potential orthogonal to the transverse magnetic field. In addition to the Poisson-like equation for the longitudinal component of the non-oscillatory problem, a general curl-curl Helmholtz equation describes the transverse problem covering both, non-oscillatory and oscillatory case. The resulting variational forms are treated by the usual nodal finite element method for the longitudinal problem and by a two-dimensional edge element method for the transverse problem. The numerical solution can be computed independently for each harmonic using fewer degrees of freedom than a three-dimensional simulation at comparable accuracy. These properties make the developed approach useful for the analysis of perturbation harmonics in magnetic plasma confinement devices in cylindrical coordinates or magnetic flux coordinates. The method is validated on analytical and numerical test cases and applied to a racetrack-shaped coil setup.

Discrete-Time State-Space Construction Method for SSO Analysis of Renewable Power Generation Integrated AC/DC Hybrid System

Construction of the linearized state-space model is the key step of eigenvalue analysis for SSO study of power systems with large-scale renewable power generation (RPG) integrated. This paper proposes a novel method to construct the state matrix of the AC/DC hybrid power system with RPG within the discrete-time domain. The continuous-time state-space model of each AC/DC component as well as RPG units in the power system, is discretized together with their corresponding control. The discretized equation of each component is then represented by an equivalent circuit consisting of parallel historical current and conductance. According to the power system configuration, the discrete-time equivalent circuit of the whole AC/DC hybrid system is established, and the nodal analysis method is applied for the construction of the discrete-time state-space model of the power system. In this way, the state matrix can be acquired much more conveniently than that of continuous-time state-space method, which needs topology analysis to find independent state variables of the network from the proper tree. The proposed method has been applied in the eigenvalue analysis for the SSO study of a case system adapted from the AC/DC hybrid transmission system in northwest China. Eigenvalue results are in high accordance with time-domain simulation results in PSCAD/EMTDC.

Towards Elastoplastic Deformation of a Solid Body by the Hybrid Godunov Method and by the Multidimensional Nodal Method of Characteristics

Towards Elastoplastic Deformation of a Solid Body by the Hybrid Godunov Method and by the Multidimensional Nodal Method of Characteristics

ISI TAŞINIMI İÇİN YAPISAL OLMAYAN AĞLARDA BİR HIZLANDIRILMIŞ SÜREKSİZ GALERKİN METDU: FORMÜLASYONU VE DOĞRULANMASI

We present a GPU-accelerated method for large scale, coupled incompressible fluid flow and heat transfer problems. A high-order, nodal discontinuous Galerkin method is utilized to discretize governing equations on unstructured triangular meshes. A semi-implicit scheme with explicit treatment of the advective terms and implicit treatment of the split Stokes operators are used for time discretization. The pressure system is solved with a conjugate gradient method together with a fully GPU-accelerated multigrid preconditioner. The code is built on scalable libParanumal solver which is a library of high-performance kernels for high-order discretizations. Performance portability is achieved by using the open concurrent compute abstraction, OCCA. A set of numerical experiments including free and mixed convection problems indicate that our approach experimentally reaches design order of accuracy.

Development of nodal diffusion code RAST-V for Vodo–Vodyanoi Energetichesky reactor analysis

This paper presents the development of a nodal diffusion code, RAST-V, and its verification and validation for VVER (vodo–vodyanoi energetichesky reactor) analysis. A VVER analytic solver has been implemented in an in-house nodal diffusion code, RAST-K. The new RAST-K version, RAST-V, uses the triangle-based polynomial expansion nodal method. The RAST-K code provides stand-alone and two-step computation modes for steady-state and transient calculations. An in-house lattice code (STREAM) with updated features for VVER analysis is also utilized in the two-step method for cross-section generation. To assess the calculation capability of the formulated analysis module, various verification and validation studies have been performed with Rostov-II, and X2 multicycles, Novovoronezh-4, and the Atomic Energy Research benchmarks. In comparing the multicycle operation, rod worth, and integrated temperature coefficients, RAST-V is found to agree with measurements with high accuracy which RMS differences of each cycle are within ±47 ppm in multicycle operations, and ±81 pcm of the rod worth of the X2 reactor. Transient calculations were also performed considering two different rod ejection scenarios. The accuracy of RAST-V was observed to be comparable to that of conventional nodal diffusion codes (DYN3D, BIPR8, and PARCS).

Investigation of Nodal Numerical Adjoints From CMFD-Based Acceleration Methods

In this work, the attribute of mathematical adjoints acquired from various CMFD (coarse-mesh finite difference) accelerated nodal methods based on the nodal expansion method (NEM) is presented. The direct transposition is implemented to the NEM-CMFD system matrix that includes correction factors to acquire the adjoint flux. Three different acceleration schemes are considered in this paper, which are, namely, CMFD, pCMFD, and one-node CMFD methods, and the self-adjoint attribute of migration operator for each acceleration scheme is studied. With regard to the one-node CMFD acceleration, a mathematical description for encountering negative adjoint flux values is given alongside an adjusted one-node CMFD scheme that stifles such an occurrence. The overall features of the aforementioned acceleration methods are recognized through analyses of 2D reactor problems including the KWU PWR 2D benchmark problem. Further systematic assessment is conducted based on the first-order perturbation theory, where the obtained adjoint fluxes are applied as weighting functions. It is clearly shown that the adjusted one-node CMFD scheme results in an improved reactivity estimation by excluding the presence of negative adjoint flux values.

Nonlinear seismic response and amplification effect of 3D sedimentary basin based on bounding surface constitutive model

Sedimentary basins can have significant amplification effects on seismic ground motion. It is very necessary to use appropriate nonlinear soil constitutive model to quantitatively analyze the seismic response of three-dimensional (3D) sedimentary basins subjected to strong ground motion. This paper employs a bounding surface constitutive model to simulate the soil nonlinearity, which is successfully incorporated into ABAQUS software. A finite element model is then established to investigate the nonlinear seismic response of 3D sedimentary employing the constitutive model along with viscoelastic artificial boundary and equivalent nodal force method. The nonlinear amplification effect of a typical 3D layered sedimentary basin is extensively evaluated. The influence of a soft interlayer in the sedimentary basin on the seismic amplification is further analyzed. The results demonstrate that the maximum surface peak acceleration usually occurs in specific areas; as the input ground motion intensity increases, the focusing and amplification effects become more significant in deeper soil layers. A soft interlayer will significantly increase the short period component of the surface response spectrum, and shorten the predominant period of sedimentary basin. These results can provide a reference basis for seismic risk analysis and determination of ground motion parameters in a sedimentary basin.

An analytical nodal method for solving the SN transport equation in 2-D Cartesian geometry

A novel Analytical Nodal Discrete Ordinates (ANDO) method for the solution of the discrete ordinates (SN) neutron transport equation in Cartesian geometry with isotropic scattering and constant neutron source is presented. A nodal method approximates the multi-dimensional transport equation as a system of coupled one-dimensional transport equations along each coordinate axis through transverse integration. The resulting transverse-integrated equations are then solved separately using a one-dimensional solution technique. The one-dimensional solution utilized in the new ANDO method is based on a recent closed-form analytical solution in slab geometry (WANG and BYAMBAAKHUU, 2017). The novel solution approach allows for a compact closed-form solution on each computational cell. Results for constant and linear transverse leakage approximations are presented. The ANDO method possesses a number of favorable properties, demonstrated with numerical results, such as high accuracy, rapid convergence, asymptotic preserving, positivity preserving, near linear computational complexity, and local-hp adaptivity. The ANDO method can also be easily extended to higher-order transverse leakage approximations, to three-dimensional Cartesian geometry, and to multi-group.

Verification of the Parallel Transport Codes Parafish and AZTRAN with the TAKEDA Benchmarks

With the increase in computational resources, parallel computation in neutron transport codes is inherent since it allows simulations with high spatial-angular resolution. Among the different methodologies available for the solution of the neutron transport equation, spherical harmonics (PN) and discrete-ordinates (SN) approximations have been widely used, as they are established classical methods for performing nuclear reactor calculations. This work focuses on describing and verifying two parallel deterministic neutron transport codes under development. The first one is the Parafish code that is based on the finite-element method and PN approximation. The second one is the AZTRAN code, based on the RTN-0 nodal method and SN approximation. The capabilities of these two codes have been tested on the TAKEDA benchmarks and the results obtained show good behavior and accuracy compared to the Monte Carlo reference solutions. Additionally, the speedup obtained by each code in the parallel execution is acceptable. In general, the results encourage further improvement in the codes to be comparable to other well-validated deterministic transport codes.

Triangular Polynomial Expansion Nodal Method for VVER Core Analysis

Due to the low computational cost, nodal diffusion methods are still commonly used to simulate full-core reactor problems. This work represents the developmental effort to build an accurate nodal kernel to treat hexagonal geometry in the core simulator code PARCS. An innovative method called TriPEN-9 has been developed by splitting a hexagonal assembly into six triangular nodes and solved using cubic polynomial expansion for the scalar flux with nine-term expansion coefficients. The nodal diffusion calculation is further accelerated with the multilevel coarse-mesh finite difference method. The verification of the TriPEN-9 method on the VVER full-core problem is provided with the model based on the NURESIM (Nuclear Reactor Simulator)-SP1 V1000-2D-C1-tr benchmark problem. The Serpent Monte Carlo code is used as a reference solution for verification and to generate homogenized group-constants data for PARCS. Exact discontinuity factors were generated in GenPMAXS, a cross-section processing code, using a similar expansion method as the TriPEN-9 core solver method with the utilization of heterogeneous solutions from Serpent. Implementing the TriPEN-9 method in PARCS, this approach can exactly reproduce the solutions from the high-fidelity Serpent calculations.

Development and verification of a three-dimensional spatial dynamics code for molten salt reactors

Molten Salt Reactor (MSR) has the characteristics of drift effect of delayed neutron precursors (DNP), direct heating of coolant and moderator cooled by fuel salt under most conditions, which bring significant differences from traditional reactors in dynamics calculation. To deal with the unique dynamics characteristics of MSR, a specific three-dimensional spatial dynamics code – TMSR3D has been developed, in which the fourth order standard nodal expansion method and the exponential transformation method are implemented to solve the neutron diffusion equations and the neutron kinetics equations respectively, the parallel multi-channel thermal hydraulics (TH) model is adopted to take into account the feedbacks of local temperature to group constants, and a convection term is introduced to the traditional balance equations for DNP to consider the influence of fuel salt flow. Besides, the least square fitting method is employed to parameterize the few-group homogenized parameters generated by the lattice code DRAGON5. The 3D-LRA core benchmark is first used to verify TMSR3D for solid-fueled reactor problem. Afterwards, the experimental data of the Molten Salt Reactor Experiment (MSRE) collected in the European project MOST are simulated, which includes the transients perturbed by protected fuel pump start-up and coast-down at zero power, and the natural circulation experiment. The numerical results indicate that the developed code is capable of providing a reasonable description for the neutronics-TH coupling phenomena of graphite-moderated channel-type MSRs.

A high-order nodal discontinuous Galerkin method for simulation of three-dimensional non-cavitating/cavitating flows

In this study, the nodal discontinuous Galerkin method is formulated in three-dimensions and applied to simulate three-dimensional non-cavitating/cavitating flows. For this aim, the three-dimensional preconditioned Navier-Stokes equations based on the artificial compressibility approach considering appropriate source terms to model cavitating phenomena are used. The spatial derivative terms in the resulting equations are discretized by utilizing the nodal discontinuous Galerkin method on tetrahedral elements and the derivative of the solution vector with respect to the artificial time is discretized by applying an explicit time integration method. An artificial viscosity method is formulated in three-dimensions and applied to properly capture the discontinuities in the cavitating flow-field. Here, different non-cavitating/cavitating flow problems including non-cavitating flow in a 3:1:1 driven cavity, non-cavitating flow between two concentric rotating spheres, cavitating flow over a sphere and cavitating flow over a wavy cylinder are solved to examine the accuracy and robustness of the proposed solution method. These flow problems are selected to properly assess the numerical method in the presence of straight and curved boundaries and for different Reynolds and cavitation numbers. From linear to cubic polynomial degrees are used in the simulations and the numerical results are compared with the available analytical and numerical results. Indications are that the present numerical method based on the NDGM formulated in three-dimensions can accurately simulate the three-dimensional non-cavitating/cavitating flows. • Formulating a nodal discontinuous Galerkin method (NDGM) in 3D to simulate noncavitating/cavitating flows. • Implementing an artificial viscosity approach in 3D for capturing the steep discontinuities in the cavity interface. • Examining the accuracy and robustness of the solution method by computing different 3D noncavitating/cavitating flow problems. • Considering the 3D NDGM proposed to assess other solution methods for computing noncavitating/cavitating flows.

Evaluation of elite sugarcane clones/varieties against red rot disease (Colletotrichum falcatum) and their suitability in crop improvement programme

Abstract. To manage red rot disease in sugarcane, several methods like use of bio-control agents, hot water and hot air treatment of setts, chemical fungicides, various cultural practices like use of disease free setts, rouging, crop rotation, irrigation management, avoiding rationing, etc. are practised. But none of the practices are able to control the disease. The most viable strategy is to manage through host plant resistance. Thus, a field experiment was conducted for two consecutive years during 2019 and 2020 at the Research Farm of Sugarcane Research Station, Bethuadahari (West Bengal), India to evaluate some elite sugarcane clones/varieties against red rot disease. The elite clones/varieties were laid out in Completely Randomized Block Design with a plot size of 2R x 6m x 0.90m with 2 replications and two red rot pathotypes Cf 07 and Cf 08 were used for artificial inoculation in plug and nodal method. The experiment revealed that the sugarcane clones CoP 16436, CoP 16437, CoP 16438, CoLk 16466, CoLk 16467, CoLk 16468, CoSe 16451, CoP 16439, CoP 16440, BO 156, CoLk 16469, CoLk 16470, CoLk 16471, CoSe 16452, CoSe 16453, CoSe 16454, CoP 17436, CoP 17437, CoP 17438, CoP 17440, CoP 17441, CoSe 17451, CoSe 16455, CoSe 16456, CoP 17444, CoP 17446 and CoSe 17452 were found as moderately resistant (MR) in plug method and resistant (R) in nodal method to both pathotypes and can be used for further evaluation in subsequent varietal developmental as well as breeding programme. The popular sugarcane varieties, namely CoLk 94184, CoSe 01421, BO 91, CoP 9301 and CoP 06436 were found as moderately resistant (MR) in plug method and resistant (R) in nodal method and can be continued for commercial cultivation as well as red rot breeding programme. As the existing popular variety CoSe 95422 was found moderately susceptible (MS) in plug method and susceptible (S) in nodal method to Cf 07 and Cf 08 red rot pathotypes, thus the variety must be discarded from commercial cultivation with immediate effect.

Conservative Finite Element Modeling of EEG and MEG on Unstructured Grids.

For interpretation of electroencephalography (EEG) and magnetoencephalography (MEG) data, multiple solutions of the respective forward problems are needed. In this paper, we assess performance of the mixed-hybrid finite element method (MHFEM) applied to EEG and MEG modeling. The method provides an approximate potential and induced currents and results in a system with a positive semi-definite matrix. The system thus can be solved with a variety of standard methods (e.g. the preconditioned conjugate gradient method). The induced currents satisfy discrete charge conservation law making the method conservative. We studied its performance on unstructured tetrahedral grids for a layered spherical head model as well as a realistic head model. We also compared its accuracy versus the conventional nodal finite element method ( P1 FEM). To avoid modeling singular sources, we completed our computations with a subtraction approach; the derived expression for the MEG response different from earlier published and involves integration of finite quantities only. We conclude that although the MHFEM is more computationally demanding than the P1 FEM, its use is justified for EEG and MEG modeling on low-resolution head models where P1 FEM loses accuracy.