scholarly journals A SIMPLIFIED TWO-NODE COARSE-MESH FINITE DIFFERENCE METHOD FOR PIN-WISE CALCULATION WITH SP3

2021 ◽  
Vol 247 ◽  
pp. 02023
Author(s):  
Wenbo Zhao ◽  
Yingrui Yu ◽  
Xiaoming Chai ◽  
Zhonghao Ning ◽  
Bin Zhang ◽  
...  
Keyword(s):  
Finite Difference ◽  
Legendre Polynomials ◽  
Coarse Mesh ◽  
Coupling Coefficients ◽  
Fine Mesh ◽  
Davidson Method ◽  
Comparable Accuracy ◽  
Current Coupling ◽  
Node Problem ◽  
3D Problem

For accurate and efficient pin-by-pin core calculation of SP3 equations, a simplified two-node Coarse Mesh Finite Difference (CMFD) method with the nonlinear iterative strategy is proposed. In this study, the two-node method is only used for discretization of Laplace operator of the 0th moment in the first equation, while the fine mesh finite difference (FMFD) is used for the 2nd moment flux and the second equation. In the two-node problem, transverse flux is expanded to second-order Legendre polynomials. In addition, the associated transverse leakage is approximated with flat distribution. Then the current coupling coefficients are updated in nonlinear iterations. The generalized eigenvalue problem from CMFD is solved using Jacobi-Davidson method. A protype code CORCA-PIN is developed. FMFD scheme is implemented in CORCA-PIN as well. The 2D KAIST 3A benchmark problem and extended 3D problem, which are cell homogenized problems with strong absorber, are tested. Numerical results show that the solution of the simplified two-node method with 1×1 mesh per cell has comparable accuracy of FMFD with 4×4 meshes per cell, but cost less time. The method is suitable for whole core pin-wise calculation.

scholarly journals An Adaptive Approach to Solutions of Fredholm Integral Equations of the Second Kind

2014 ◽  
Vol 2014 ◽  
pp. 1-13
Author(s):  
Nebiye Korkmaz ◽  
Zekeriya Güney
Keyword(s):  
Galerkin Method ◽  
Integral Equations ◽  
Iteration Method ◽  
Legendre Polynomials ◽  
Optimization Problem ◽  
Approximate Solutions ◽  
Basis Functions ◽  
Fredholm Integral Equations ◽  
Coarse Mesh ◽  
Fine Mesh

As an approach to approximate solutions of Fredholm integral equations of the second kind, adaptive hp-refinement is used firstly together with Galerkin method and with Sloan iteration method which is applied to Galerkin method solution. The linear hat functions and modified integrated Legendre polynomials are used as basis functions for the approximations. The most appropriate refinement is determined by an optimization problem given by Demkowicz, 2007. During the calculationsL2-projections of approximate solutions on four different meshes which could occur between coarse mesh and fine mesh are calculated. Depending on the error values, these procedures could be repeated consecutively or different meshes could be used in order to decrease the error values.

Controlling Numerical Dispersion by Variably Timed Flux Updating in Two Dimensions

1982 ◽  
Vol 22 (03) ◽  
pp. 409-419 ◽  
Author(s):  
R.G. Larson
Keyword(s):  
Finite Difference ◽  
Pressure Gradient ◽  
Single Point ◽  
Companion Paper ◽  
Two Dimensions ◽  
Independent Component ◽  
Numerical Dispersion ◽  
Coarse Mesh ◽  
Fixed Concentration ◽  
Order Of Magnitude

Abstract The variably-timed flux updating (VTU) finite difference technique is extended to two dimensions. VTU simulations of miscible floods on a repeated five-spot pattern are compared with exact solutions and with solutions obtained by front tracking. It is found that for neutral and favorable mobility ratios. VTU gives accurate results even on a coarse mesh and reduces numerical dispersion by a factor of 10 or more over the level generated by conventional single-point (SP) upstream weighting. For highly unfavorable mobility ratios, VTU reduces numerical dispersion. but on a coarse mesh the simulation is nevertheless inaccurate because of the inherent inadequacy of the finite-difference estimation of the flow field. Introduction A companion paper (see Pages 399-408) introduced the one-dimensional version of VTU for controlling numerical dispersion in finite-difference simulation of displacements in porous media. For linear and nonlinear, one- and two-independent-component problems, VTU resulted in more than an order-of-magnitude reduction in numerical dispersion over conventional explicit. SP upstream-weighted simulations with the same number of gridblocks. In this paper, the technique is extended to two dimensional (2D) problems, which require solution of a set of coupled partial differential equations that express conservation of material components-i.e., (1) and (2) Fi, the fractional flux of component i, is a function of the set of s - 1 independent-component fractional concentrations {Ci}, which prevail at the given position and time., the dispersion flux, is given by an expression that is linear in the specie concentration gradients. The velocity, is proportional to the pressure gradient,. (3) where lambda, in general, can be a function of composition and of the magnitude of the pressure gradient. The premises on which Eqs. 1 through 3 rest are stated in the companion paper. VTU in Two Dimensions The basic idea of variably-timed flux updating is to use finite-difference discretization of time and space, but to update the flux of a component not every timestep, but with a frequency determined by the corresponding concentration velocity -i.e., the velocity of propagation of fixed concentration of that component. The concentration velocity is a function of time and position. In the formulation described here, the convected flux is upstream-weighted, and all variables except pressure are evaluated explicitly. As described in the companion paper (SPE 8027), the crux of the method is the estimation of the number of timesteps required for a fixed concentration to traverse from an inflow to an outflow face of a gridblock. This task is simpler in one dimension, where there is only one inflow and one outflow face per gridblock, than it is in two dimensions, where each gridblock has in general multiple inflow and outflow faces. SPEJ P. 409^

scholarly journals Analytic Coarse-Mesh Finite-Difference Method Generalized for Heterogeneous Multidimensional Two-Group Diffusion Calculations

2003 ◽  
Vol 144 (1) ◽  
pp. 23-35 ◽  
Author(s):  
Nuria García-Herranz ◽  
Oscar Cabellos ◽  
José M. Aragonés ◽  
Carol Ahnert
Keyword(s):  
Finite Difference ◽  
Finite Difference Method ◽  
Difference Method ◽  
Coarse Mesh

scholarly journals Discontinuity Capture in One-Dimensional Space Using the Numerical Manifold Method with High-Order Legendre Polynomials

2020 ◽  
Vol 10 (24) ◽  
pp. 9123
Author(s):  
Yan Zeng ◽  
Hong Zheng ◽  
Chunguang Li
Keyword(s):  
Finite Difference ◽  
Finite Difference Method ◽  
Finite Volume Method ◽  
Difference Method ◽  
Legendre Polynomials ◽  
Dimensional Space ◽  
Numerical Manifold Method ◽  
One Dimensional ◽  
Numerical Manifold ◽  
Volume Method

Traditional methods such as the finite difference method, the finite element method, and the finite volume method are all based on continuous interpolation. In general, if discontinuity occurred, the calculation result would show low accuracy and poor stability. In this paper, the numerical manifold method is used to capture numerical discontinuities, in a one-dimensional space. It is verified that the high-degree Legendre polynomials can be selected as the local approximation without leading to linear dependency, a notorious “nail” issue in Numerical Manifold Method. A series of numerical tests are carried out to evaluate the performance of the proposed method, suggesting that the accuracy by the numerical manifold method is higher than that by the later finite difference method and finite volume method using the same number of unknowns.

scholarly journals Detecting radiolaria in the field

1991 ◽  
Vol 10 (1) ◽  
pp. 22-22 ◽  
Author(s):  
Simon K. Haslett ◽  
Paul D. Robinson
Keyword(s):  
Rock Type ◽  
Fine Fraction ◽  
Field Work ◽  
Coarse Mesh ◽  
Extraction Techniques ◽  
The Past ◽  
Fine Mesh ◽  
Rock Types ◽  
Siliceous Rock ◽  
Calcareous Microfossils

Abstract. Radiolaria can be preserved in all types of marine sedimentary rocks, the method for their extraction being dependent on the mineralogy of the radiolarian test and the nature of the rock-type in which they occur. In the past radiolaria could only be viewed in thin section (Hinde, 1890; Hinde & Fox, 1895), with no method of detecting the presence of radiolaria prior to sectioning. Modern extraction techniques are normally laboratory based and use hazardous chemicals, therefore it is advantageous to establish the radiolarian content of the sample before collection and transportation back to the laboratory. This can be achieved in a number of ways:-1. Non-lithified sediments. Radiolaria are separated from the sediment by washing the sample over a set of small sieves. Two mesh sizes should be used, a coarse mesh around 150μm to separate large litho-fragments, and a fine mesh no greater than 63μm to concentrate the radiolaria. The fine fraction is then washed with dilute hydrochloric acid (HCl) to eliminate the calcareous microfossils, leaving a pure radiolarian sludge, which is dried on filter paper.2. Siliceous rock-types. Methods for extracting radiolaria from cherts have been in use since the early 1970’s (Dumitrica, 1970; Pessagno & Newport, 1972), and have recently been applied to field-work (Cordey & Krauss, 1990). The recognition of fossiliferous bedded cherts is possible with the use of a hand-lens in good sunlight. If radiolaria are present, they should be detectable as small protrusions, especially along laminae. To extract the radiolaria, break up the sample. . .

Lobatto Point Quadrature for Thermal Lubrication Problems Involving Compressible Lubricants

2005 ◽  
Author(s):  
L. Moraru ◽  
T. G. Keith
Keyword(s):  
Calculation Method ◽  
Efficient Algorithm ◽  
Legendre Polynomials ◽  
Continuity Equation ◽  
Transverse Velocity ◽  
Fine Mesh ◽  
Point Calculation ◽  
Temperature And Pressure ◽  
Grid Points

Refined solutions of thermal lubrication problems generally require fine mesh and many iteration steps. To resolve these difficulties, Elrod and Brewe (1) proposed an efficient algorithm based on the use of Lobatto point quadrature. Within this approach, the unknown temperature across the film is written in a series of Legendre polynomials. This paper presents a Lobatto point quadrature algorithm which is applicable for thermal lubrication problems with compressible lubricants. In this case both density and viscosity of the lubricant are taken to be temperature and pressure dependent. The transverse velocity is obtained from the continuity equation. Use of the Labatto point calculation method has resulted in greater accuracy without the use of a large number of grid points.

scholarly journals A Comparison of Splitting Techniques for 3D Complex Padé Fourier Finite Difference Migration

2011 ◽  
Vol 2011 ◽  
pp. 1-12 ◽  
Author(s):  
Jessé C. Costa ◽  
Débora Mondini ◽  
Jörg Schleicher ◽  
Amélia Novais
Keyword(s):  
Wave Equation ◽  
Finite Difference ◽  
Computational Cost ◽  
Three Dimensional ◽  
Depth Level ◽  
Numerical Examples ◽  
The Cost ◽  
Comparable Quality ◽  
3D Problem ◽  
Dimensional Wave

Three-dimensional wave-equation migration techniques are still quite expensive because of the huge matrices that need to be inverted. Several techniques have been proposed to reduce this cost by splitting the full 3D problem into a sequence of 2D problems. We compare the performance of splitting techniques for stable 3D Fourier finite-difference (FFD) migration techniques in terms of image quality and computational cost. The FFD methods are complex Padé FFD and FFD plus interpolation, and the compared splitting techniques are two- and four-way splitting as well as alternating four-way splitting, that is, splitting into the coordinate directions at one depth and the diagonal directions at the next depth level. From numerical examples in homogeneous and inhomogeneous media, we conclude that, though theoretically less accurate, alternate four-way splitting yields results of comparable quality as full four-way splitting at the cost of two-way splitting.

Modelling and Simulation of Temperature and Stress Analysis in TIG Welding Process

2015 ◽  
Vol 830-831 ◽  
pp. 294-297
Author(s):  
Nayan Chandak ◽  
Mohan Kumar Pradhan ◽  
Lokesh Boriwal
Keyword(s):  
Residual Stress ◽  
Structural Analysis ◽  
Strain State ◽  
Processing Time ◽  
Welding Process ◽  
Time Interval ◽  
Coarse Mesh ◽  
Ansys Software ◽  
Fine Mesh ◽  
Material Conditions

In this study, the welding process is modelled and analysed using ANSYS software. The temperature and residual stress produced during the process is depicted. During heating, the material conditions, parts affected by residual stress and the stress–strain state at different time interval is recorded and a subsequent structural analysis is used for the analysis, the same is used in the analysis where thermal and structural results are investigated. Subsequently, with sensitivity analysis the results are evaluated. Non-uniform meshing is used to entrap the result with fine mesh in the heat affected zone and coarse mesh away from it to save processing time. The results from the thermal structural analysis are presented to understand the process deeply and comparison of the graph plot between temperature and time is explained.

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