Nodal Method Research Articles

A modified cell-centered nodal integral scheme for the convection-diffusion equation

The nodal integral methods (NIMs) are very efficient and accurate coarse-mesh methods for solving partial differential equations. The cell-centered NIM (CCNIM) is a simplified variant of the NIMs that has recently shown its efficiency in solving fluid flow problems but has been hampered by issues such as inapplicability to one-dimensional problems, complexities in handling Neumann boundary conditions and the formulation of a system of differential-algebraic equations (DAEs) for discrete unknowns. Here, we present a modified version of the CCNIM designed to overcome the challenges associated with its previous version. Our novel development retains the essence of CCNIM while resolving these issues. The proposed scheme, grounded in the nodal framework, achieves second-order accuracy in both spatial and temporal dimensions. Unlike its precursor, the proposed method formulates algebraic equations for discrete variables per node, eliminating the cumbersome DAE system. Neumann boundary conditions are seamlessly incorporated through a straightforward flux definition, and applicability to one-dimensional problems is now feasible. We successfully apply our approach to one and two-dimensional convection-diffusion problems with known analytical solutions to validate our approach. The simplicity and robustness of the approach lay the foundation for its seamless extension to more complex fluid flow problems.

Nonlinear motion modeling and analysis of underwater cables under complex loads based on global nodal position finite element method in Hamiltonian formulation

The nonlinear dynamic response of underwater cable is essential for underwater equipment system. This paper presents an effective global nodal position finite element method in Hamiltonian formulation to investigate the nonlinear motion of underwater cable under complex loads. The method fully considers the axial, bending, torsional and shear deformation of underwater cable. The non-linear Green-Lagrange strain-displacement relationship is applied to obtain the strain energy. Furthermore, the fourth-order circular cross-section quadrature is introduced to integrate the volume of underwater cable element after deformation. The dynamic governing equations in Hamiltonian formulation are derived and converted into the corresponding canonical equations, which are calculated by the symplectic difference algorithm. Then an initial curved pendulum system, a 180° turn towed underwater cable system and the spiral turning motion of ROV are designed to verify the presented method. The results show that compared with the existing Hamiltonian global nodal position finite element method which ignores torsional and shear deformation, the proposed method is more accurate in describing the nonlinear motion of the underwater cable with axial, bending, torsional and shear deformation.

Simulating Fuel Assembly Bowing: A Neutron-Diffusion Method Based on Arbitrary Quadrilateral Node and Conformal Mapping Technique

Fuel assembly bowing, widely observed in a pressurized water reactor (PWR), often results in an asymmetrical power distribution. This paper proposes a neutron-diffusion method that integrates the arbitrary quadrilateral node with the conformal mapping technique to characterize the impact of fuel assembly bowing on power distribution. The proposed method involves a nonlinear iteration process to solve the neutron-diffusion equation. The global coarse-mesh finite difference equation is established on the arbitrary quadrilateral nodes, which are redivided in response to fuel assembly bowing. The local two-node nodal expansion method equation is established on the rectangular nodes, which are mapped from the original arbitrary quadrilateral nodes using the conformal mapping technique. The proposed method has improved our self-developed core code, named SPARK, for PWRs. To verify this novel method, two distinct types of fuel assembly bowing are modeled based on the mini core. The reference results for these models were obtained using the Monte Carlo code NECP-MCX. The numerical results suggest a robust agreement between the biases of keff and power distributions and their corresponding reference results.

Numerical modelling of downward progressive landslides in sensitive clay

Landslides in sensitive clays are catastrophic events that threaten life and property. When plastic strain occurs in sensitive clays, there is a significant reduction in the undrained shear strength. Consequently, even minor triggers can lead to multiple progressive landslides, causing extensive devastation. Investigating landslides in sensitive clay presents challenges, necessitating robust numerical methods to comprehend failure modes and collapse behaviors, particularly in the post-failure phase. Although several numerical methods have been utilized for simulations, most have primarily focused on upward progressive landslides. In this study, we adopted the nodal integration-based particle finite element method to simulate downward progressive landslides triggered by surcharge loads from an embankment near the crest in sensitive clay on a long slope. We also considered nonlinear strain softening in sensitive clays. The effects of the strain softening rate and strength gradient of the sensitive clay layer on the failure mechanisms and destruction extent were investigated. The simulation results reveal that a downward progressive landslide initiates with a localized flow slide surrounding the embankment and usually involves a combination of various failure modes.

VINUS: A neutron transport solver based on the variational nodal method for reactor core analysis

Compared to traditional transverse integration methods, the variational nodal method, with its unique advantages, is more suitable for high-fidelity calculations of reactor physics in reactors with complex geometries and finer detail descriptions. In this study, the basic theory of the variational nodal method was derived and the VINUS code is developed. The neutron solver based on this method is adaptable to various geometric models, and showcased the code's fundamental framework. On this basis, a set of self-designed macroscopic cross-section benchmarks, actual macroscopic cross-section benchmark VVER-440, and few-group microscopic cross-section benchmark RBEC-M for fast spectrum reactors were used to verify different functionalities of VINUS. The results were shown that VINUS code maintains good computational accuracy and convergence trends. For the VVER-440 benchmarks, the deviation of keff of VINUS from reference is less than 100 pcm, and the maximum power deviation is less than 4 %. For the RBEC-M, the deviation of keff is 125 pcm, and the maximum power deviation is less than 5 %. These outcomes collectively demonstrate the solver's potential for engineering applications in future advanced reactor designs.

A nodal discontinuous Galerkin method for wave propagation in coupled acoustic–elastic media

AbstractThe accurate numerical solution at an acoustic–elastic interface is important for offshore exploration. The solution requires careful implementation for the acoustic–elastic boundary conditions. In this work, we leverage a nodal discontinuous Galerkin method, in which the unstructured uniform triangular meshes are used for the model meshing and an explicit upwind numerical flux derived from the Riemann problem is adopted to handle the boundary conditions at the acoustic–elastic interface. Several numerical results are provided to assess the accuracy and convergence properties of this method. The convergence analysis is carried out in the coupled model with a flat interface, and the accuracy of the proposed method is verified in the curved interface coupled model. Finally, a more complex model with a salt dome, inspired by real geophysical applications, is carried out in this study. The numerical results demonstrate that the proposed nodal discontinuous Galerkin method is effective and accurate for dealing with the coupled acoustic–elastic media with complex geometries.

Research on pin-by-pin calculation method of rectangular mesh based on quasi-diffusion theory

With the development of nuclear energy techniques, pin-by-pin calculation using the SP3 method based on pin cell homogenization attracts researchers. This study proposed a pin-by-pin calculation method of rectangular mesh based on quasi-diffusion (QD) theory. The QD equation and boundary condition were derived from a transport equation similar to the P1 equation derivation. The Eddington factor was treated as a few-group homogenized parameter and calculated based on a deterministic code ALPHA. The numerical method of QD equation was developed based on nodal expansion method of nonlinear iteration. Some benchmarks were used to verify the Eddington factor and the numerical method of QD. 4 colorset models with different neutron anisotropy levels were analyzed by pin-by-pin-QD method. Pin-by-pin-QD can reduce keff error by about 600 pcm and agree better with reference results. Additionally, it has better accuracy and similar efficiency as diffusion calculation and is suitable for problems with strong anisotropy.

An improved and simplified approach for the steady-state analysis of self-excited induction generators using binary search algorithm

PurposeThis paper aims at developing an improved method, based on binary search algorithm (BSA) for the steady-state analysis of self-excited induction generators (SEIGs), which are increasingly used in wind energy electric conversion systems. The BSA is also compared with linear search algorithm (LSA) to bring out the merits of BSA over LSA.Design/methodology/approachAll the parameters of SEIG, including the varying core loss of the machine, have been considered to ensure accuracy in the predetermined performance values of the set up. The nodal admittance method has been adopted to simplify the equivalent circuit of the generator and load. The logic and steps involved in the formulation of the complete procedure have been illustrated using elaborate flowcharts.FindingsThe proposed approach is validated by the experimental results, obtained on a three-phase 240 V, 5.0 A, 2.0 kW SEIG, which closely match with the corresponding predicted performance values. The analysis is shown to be easy to implement with reduced computation time.Originality/valueA novel improved and simplified technique has been formulated for estimating the per unit frequency (a), magnetizing reactance (Xm) and core loss resistance (Rm) of the SEIG using the nodal admittance of its equivalent circuit. The accuracy of the predetermined performance is enhanced by considering the SEIG’s varying core loss. Only simple MATLAB programming has been used for adopting the algorithms.

An efficient parallel algorithm of variational nodal method for heterogeneous neutron transport problems

An efficient parallel algorithm of variational nodal method for heterogeneous neutron transport problems

Production Optimization of an Oil Well by Restraining Water Breakthrough

This study investigates the well named X (for confidential reasons) of the field called Y which initially was productive with the natural energy of the reservoir of the oil in the absence of water. After a few years of production, water began to overflow excessively in the well. The goal of this paper is to maximize the oil production in an oil well X by reducing water ingress. The Pressure Volume Temperature (PVT) data, completion data, and reservoir data are analyzed via PIPESIM and Excel software by using the nodal analysis method to get the well performance and decline curve for predictions. Two scenarios are considered: firstly, to install an electric submersible pump (ESP) to activate the X well and secondly to make a new perforation. The ESP is installed at 11300 ft where the water production flow rate is 5586.264 STB/d and the oil production flow rate is 1396.566 STB/d. The new perforation is installed at 12038 ft where the water production flow rate is 277.1693 STB/d and the oil production flow rate is 5543.387 STB/d. To have the optimal parameters, the sensitivity analysis is applied to the flowline diameter and the wellhead pressure. The optimal parameter values obtained are 308.6128 STB/d for the water production flow rate and 5863.643 STB/d for the oil production flow rate. The new perforation is appropriate because this scenario allows water reduction, oil production maximization, profitability of 98086854 $, and a return on investment in 5 months during 16 years of production.

A leap-frog nodal discontinuous Galerkin method for Maxwell polynomial chaos Debye model

The Maxwell polynomial chaos Debye model uses the parameter distribution in the Debye model to represent the Cole-Cole dispersive mechanism. In this paper, a leap-frog nodal discontinuous Galerkin (DG) method is studied for solving this model. Under the Courant-Friedrichs-Lewy (CFL) condition, the stability of the fully discrete DG scheme is demonstrated and its error estimate is derived. Numerical examples are provided for supporting the correctness of the theoretical analysis and the effectiveness of our proposed method.

Monolithic Convex Limiting for Legendre-Gauss-Lobatto Discontinuous Galerkin Spectral-Element Methods

We extend the monolithic convex limiting (MCL) methodology to nodal discontinuous Galerkin spectral-element methods (DGSEMS). The use of Legendre-Gauss-Lobatto (LGL) quadrature endows collocated DGSEM space discretizations of nonlinear hyperbolic problems with properties that greatly simplify the design of invariant domain-preserving high-resolution schemes. Compared to many other continuous and discontinuous Galerkin method variants, a particular advantage of the LGL spectral operator is the availability of a natural decomposition into a compatible subcell flux discretization. Representing a high-order spatial semi-discretization in terms of intermediate states, we perform flux limiting in a manner that keeps these states and the results of Runge-Kutta stages in convex invariant domains. In addition, local bounds may be imposed on scalar quantities of interest. In contrast to limiting approaches based on predictor-corrector algorithms, our MCL procedure for LGL-DGSEM yields nonlinear flux approximations that are independent of the time-step size and can be further modified to enforce entropy stability. To demonstrate the robustness of MCL/DGSEM schemes for the compressible Euler equations, we run simulations for challenging setups featuring strong shocks, steep density gradients, and vortex dominated flows.

Fast production and water-breakthrough analysis methods demonstrated using Volve Field data

When producing from conventional fields, the well rates are primarily constrained by the production-system in the early years of the field-life, while later in the field-life the production rates are primarily constrained by the reservoir deliverability. For the post-plateau production period, the reservoir deliverability will no longer potentially exceed the production-system well-rate constraints. Traditionally, analytical equations are used in a nodal analysis method that balances the pressure at the well inflow point from the reservoir (inflow performance relationship; IPR) with the pressure required for the vertical lift performance (VLP; or vertical flow performance; VFP) from the same point upward. A faster and simpler approach is proposed in the present study. Whereas, the classical IPR solutions are based on a constant well-rate solution of the diffusivity equation, use of a constant bottomhole pressure assumption can bypass the need for nodal analysis type pressure matching solutions to obtain the well rate. Instead, the well rate can be directly computed from the pressure decline in the reservoir and any production system capacity constraint can be imposed on the theoretical well rate due to the reservoir quality. The merits of the new approach are explained and illustrated by way of a detailed production analysis case study using open-access data from the Volve Field (Norwegian Continental Shelf). In addition, the case study of the Volve Field wells demonstrates a new water-breakthrough analysis method.

Nodal integral method to solve the two-dimensional, time-dependent, incompressible Navier-Stokes equations in curvilinear coordinates

The numerical solution of the two–dimensional, time–dependent, incompressible Navier–Stokes equations (NSE) in curvilinear coordinates using the nodal integral method (NIM) is presented in this paper. The developed numerical scheme is applied to solve fluid flow problems in domains discretized using quadrilateral cells. Three test problems are numerically solved to assess the accuracy and efficiency of the method. The scheme is second-order accurate in space for all considered deformation levels and Re numbers. The results of the current scheme are compared with the other numerical results in the literature, and good agreement is obtained for all considered cases. The NIM demonstrates superior accuracy per cell compared to other second–order finite–volume schemes considered in this work. New benchmark results obtained from the solution on fine meshes are presented.

Development and validation of transient analysis module in nodal diffusion code RAST-V with Kalinin-3 coolant transient benchmark

This study introduces a transient analysis module developed for RAST-V and validates it using the Kalinin-3 benchmark problem. For the benchmark analysis, RAST-V standalone and STREAM/RAST-V calculations were performed. STREAM supplies the few-group constants and RAST-V conducts a 3D core simulation utilizing few-group cross-sectional data. To improve accuracy, the main solver was developed based on the advanced semi-analytic nodal method. To evaluate the computational capability of the transient analysis module in RAST-V, Kalinin-3 benchmark is employed. Kalinin-3 represents a coolant transient benchmark that offers experimental data during the deactivation of the Main Circulation Pumps. Consequently, the transient calculations reflected the changes in the reactor flow rate. Benchmark comprising steady-state and transient calculations. During the steady state, the STREAM/RAST-V combination demonstrated a 30 ppm root mean square difference from 0 to 128.50 EFPD. For the transient calculations, STREAM/RAST-V showed power differences within ±7 % over a range of 0–300 s. Axial offset differences were within ±3 %, and the RMS difference in radial power ranged within 2.596 % at both 0 and 300 s. Overall, this study effectively demonstrated the newly developed transient solver in RAST-V and validated it using the Kalinin-3 benchmark problem.

A stable implicit nodal integration-based particle finite element method (N-PFEM) for modelling saturated soil dynamics

In this study, we present a novel nodal integration-based particle finite element method (N-PFEM) designed for the dynamic analysis of saturated soils. Our approach incorporates the nodal integration technique into a generalised Hellinger-Reissner (HR) variational principle, creating an implicit PFEM formulation. To mitigate the volumetric locking issue in low-order elements, we employ a node-based strain smoothing technique. By discretising field variables at the centre of smoothing cells, we achieve nodal integration over cells, eliminating the need for sophisticated mapping operations after re-meshing in the PFEM. We express the discretised governing equations as a min-max optimisation problem, which is further reformulated as a standard second-order cone programming (SOCP) problem. Stresses, pore water pressure, and displacements are simultaneously determined using the advanced primal-dual interior point method. Consequently, our numerical model offers improved accuracy for stresses and pore water pressure compared to the displacement-based PFEM formulation. Numerical experiments demonstrate that the N-PFEM efficiently captures both transient and long-term hydro-mechanical behaviour of saturated soils with high accuracy, obviating the need for stabilisation or regularisation techniques commonly employed in other nodal integration-based PFEM approaches. This work holds significant implications for the development of robust and accurate numerical tools for studying saturated soil dynamics.

Ecological network optimization of Jiuquan City, Gansu, China based on complex network theory and circuit model

Building a scientific and reasonable ecological network is the key for optimizing the pattern of territorial development and protection, and is of great significance for ensuring regional ecological security and promoting the virtuous cycle of ecosystems. In previous studies, nodal attack method (destruction of ecological source area) was often used in the "robustness" evaluation of ecological networks. Actually, the ecological corridor is more fragile than the source area, and thus the nodal attack method is not reasonable. In this study, taking Jiuquan City as the research area, based on the circuit model to construct the ecological network, we carried out the topology optimization of ecological network by using three strategies (random edge increase, node degree and priority edge increase with low node intermedium number) in complex network theory. We compared and analyzed the "robustness" of ecological network before and after optimization by constructing edge attack strategy, and selected the best network optimization strategy. The results showed that 65 ecological source areas were identified in Jiuquan City, with a total area of 20275.15 km2, and that grassland accounted for 89.5% of the source area. We identified 179 ecological corridors with a total length of 6387.16 km, 158 ecological barrier points with a total area of 1385.5 km2. The unused land accounted for 92.2% of the total barrier points area. We identified 63 ecological pinch points, mainly concentrated in the source edge and corridor intersection. Among them, the spatial distribution of 11 barrier points and pinch points was consistent, which was the key area to be repaired in ecological network optimization. The three optimization strategies had significantly improved the stability of ecological network in Jiuquan City. The relative size of the maximum connected subgraph and the edge connected rate of the ecological network of the optimization strategy of adding edges according to degree were all the most stable under random attack mode and deliberate attack mode, which was the best optimization scheme for ecological network in Jiuquan City.

Exponential Response Matrix Spectral Nodal Method for the Discrete Ordinates Neutral Particle Transport Model

In this work, the response matrix‐exponential nodal method is presented to solve fixed source neutral particle transport problems with isotropic scattering and discrete ordinates formulation in two‐dimensional Cartesian geometry. The method is based on the response matrix scheme by using the general solution of the transverse integrated SN nodal equations with exponential approximations for the leakage terms. This approach eliminates the need for additional auxiliary equations found in the spectral Green’s function (SGF) schemes, streamlining internodal computation to a matrix‐vector multiplication. The formulation includes a lambda parameter possible determined by the medium characteristics, e.g., the scattering ratio coefficient and/or empirical methods. Two model problems with low absorption rates were investigated to analyze the performance of the proposed method. The numerical solutions exhibited superior accuracy for the coarsest spatial grid configurations in both homogeneous and heterogeneous study cases. However, refining the spatial grid revealed a relative loss in accuracy compared to the RM‐CN method, potentially due to increased truncation error propagation. Nevertheless, all numerical solutions converge to the reference solution when refining the spatial discretization grid. Comparison of both response matrix schemes demonstrated similar convergence rates and computational efficiency in terms of iterations and CPU consumption.

AUTOMATING COMPUTATIONS FOR ELECTRIC CIRCUIT ANALYSIS WITH LABVIEW

The article considers the application of the theory of the complex variable function employing the LabVIEW graphical programming environment for automating the calculation of electric circuits. The software products have been created to implement the method of constructing the matrix of the system of linear algebraic equations by inspection for both the mesh current method and nodal potential method in electric circuits of direct and alternating current and voltage using a graphical programming environment, except for circuits with dependent sources. All developed programs can be converted into .exe files, with subsequent utilization on computers that do not have a program environment installed. These software solutions can help lecturers quickly generate solutions for new assignments, saving valuable time. Students can utilize these software products for self-control, further reducing their workload.

Numerical qualification of the GSP3(0) theory using the analytic function expansion nodal method

The simplified P3 (SP3) method is an important way to implement the pin-by-pin neutron transport calculation of the core. This method has the high accuracy but costs twice the computing time of the diffusion method. Recently, the Generalized SPN (GSPN) theory has been developed, and the ad hoc SPN theory is resolved through rigorous theoretical derivation. GSP3(0) theory is the 3rd order and the 0th layer of GSPN theory. Compared with GSP3(0) theory, the SP3 theory with Marshak boundary condition is regarded as the traditional SP3 theory. The equations of the two theories are the same while the interface and the boundary conditions are different. Based on the GSP3(0) theory and the traditional SP3 theory, transport calculation codes have been developed using the analytical function expansion nodal (AFEN) method. The numerical results show that both of the two SP3 theories are more accurate than the diffusion theory, and the GSP3(0) theory can further improve the accuracy of the pin power calculation compared to the traditional SP3 theory.