scholarly journals A Revisit to CMFD Schemes: Fourier Analysis and Enhancement

Energies ◽  
2021 ◽  
Vol 14 (2) ◽  
pp. 424
Author(s):  
Dean Wang ◽  
Zuolong Zhu
Keyword(s):  
Fourier Analysis ◽  
Eigenvalue Problems ◽  
Neutron Transport ◽  
Coarse Mesh ◽  
Successive Overrelaxation ◽  
Computational Cell ◽  
Artificial Diffusion ◽  
Acceleration Method ◽  
The Stability ◽  
Partial Current

The coarse-mesh finite difference (CMFD) scheme is a very effective nonlinear diffusion acceleration method for neutron transport calculations. CMFD can become unstable and fail to converge when the computational cell optical thickness is relatively large in k-eigenvalue problems or diffusive fixed-source problems. Some variants and fixups have been developed to enhance the stability of CMFD, including the partial current-based CMFD (pCMFD), optimally diffusive CMFD (odCMFD), and linear prolongation-based CMFD (lpCMFD). Linearized Fourier analysis has proven to be a very reliable and accurate tool to investigate the convergence rate and stability of such coupled high-order transport/low-order diffusion iterative schemes. It is shown in this paper that the use of different transport solvers in Fourier analysis may have some potential implications on the development of stabilizing techniques, which is exemplified by the odCMFD scheme. A modification to the artificial diffusion coefficients of odCMFD is proposed to improve its stability. In addition, two explicit expressions are presented to calculate local optimal successive overrelaxation (SOR) factors for lpCMFD to further enhance its acceleration performance for fixed-source problems and k-eigenvalue problems, respectively.

scholarly journals DEMONSTRATION OF A LINEAR PROLONGATION CMFD METHOD ON MOC

2021 ◽  
Vol 247 ◽  
pp. 03006
Author(s):  
Jin Li ◽  
Yunlin Xu ◽  
Dean Wang ◽  
Qicang Shen ◽  
Brendan Kochunas ◽  
...  
Keyword(s):  
Neutron Transport ◽  
Coarse Mesh ◽  
Scalar Flux ◽  
Scalar Fluxes ◽  
Transport Calculation ◽  
Discontinuity Problem ◽  
Neutron Transport Equations ◽  
The Stability ◽  
Partial Current ◽  
Better Than

Coarse Mesh Finite Difference (CMFD) method is a very effective method to accelerate the iterations for neutron transport calculation. But it can degrade and even fail when the optical thickness of the mesh becomes large. Therefore several methods, including partial current-based CMFD (pCMFD) and optimally diffusive CMFD (odCMFD), have been proposed to stabilize the conventional CMFD method. Recently, a category of “higherorder” prolongation CMFD (hpCMFD) methods was proposed to use both the local and neighboring coarse mesh fluxes to update the fine cell flux, which can solve the fine cell scalar flux discontinuity problem between the fine cells at the bounary of the coarse mesh. One of the hpCMFD methods, refered as lpCMFD, was proposed to use a linear prolongation to update the fine cell scalar fluxes. Method of Characteristics (MOC) is a very popular method to solve neutron transport equations. In this paper, lpCMFD is applied on the MOC code MPACT for a variety of fine meshes. A track-based centroids calculation method is introduced to find the centroids coordinates for random shapes of fine cells. And the numerical results of a 2D C5G7 problem are provided to demonstrate the stability and efficiency of lpCMFD method on MOC. It shows that lpCMFD can stabilize the CMFD iterations in MOC method effectively and lpCMFD method performs better than odCMFD on reducing the outer MOC iterations.

scholarly journals Remarks on the Systems of Semilinear Fractional Rayleigh-Stokes Equation

2021 ◽  
Vol 2021 ◽  
pp. 1-9
Author(s):  
Le Dinh Long
Keyword(s):  
Cauchy Problem ◽  
Fourier Analysis ◽  
Stokes Equation ◽  
Stokes Equations ◽  
Newtonian Fluids ◽  
System Of Equations ◽  
Main Technique ◽  
The Cauchy Problem ◽  
The Stability

In this paper, we study the Cauchy problem for a system of Rayleigh-Stokes equations. In this system of equations, we use derivatives in the classical Riemann-Liouville sense. This system has many applications in some non-Newtonian fluids. We obtained results for the existence, uniqueness, and frequency of the solution. We discuss the stability of the solutions and find the solution spaces. Our main technique is to use the Banach mapping theorem combined with some techniques in Fourier analysis.

scholarly journals A Rayleigh quotient method for criticality eigenvalue problems in neutron transport

2020 ◽  
Vol 138 ◽  
pp. 107120
Author(s):  
M.I. Ortega ◽  
R.N. Slaybaugh ◽  
P.N. Brown ◽  
T.S. Bailey ◽  
B. Chang
Keyword(s):  
Eigenvalue Problems ◽  
Neutron Transport ◽  
Rayleigh Quotient

scholarly journals An Accelerating Convergence Rate Method for Moving Morphable Components

2020 ◽  
Vol 2020 ◽  
pp. 1-15
Author(s):  
Ruichao Lian ◽  
Shikai Jing ◽  
Zhijun He ◽  
Zefang Shi ◽  
Guohua Song
Keyword(s):  
Convergence Rate ◽  
Three Dimensional ◽  
Structural Topology ◽  
Coarse Mesh ◽  
Starting Point ◽  
Good Starting Point ◽  
Topological Configuration ◽  
Rate Method ◽  
The Stability ◽  
Moving Morphable Components

In the structural topology optimization approaches, the Moving Morphable Components (MMC) is a new method to obtain the optimized structural topologies by optimizing shapes, sizes, and locations of components. However, the size of the mesh has a strong influence on the rate of which the component builds the initial topological configuration by moving. The influence may slow down the convergence rate. In this paper, a hierarchical mesh subdivision solution method that can accelerate the convergence rate for the MMC is developed. First, the coarse mesh is used as the starting point for the optimization problem, and the construction process of the initial topology structure is increased speed by accelerating the movement of components. Second, the optimized solution obtained by the coarse mesh is equivalently mapped to the same problem with a finer mesh and used to construct a good starting point for the next optimization. Finally, two-dimensional (2D) MBB beam example and three-dimensional (3D) short cantilever beam example are provided so as to validate that with the use of the proposed approach, demonstrating that this method can improve the convergence rate and the stability of the MMC method.

scholarly journals Almost diagonal systems in asymptotic integration

1985 ◽  
Vol 28 (2) ◽  
pp. 143-158 ◽  
Author(s):  
H. Gingold
Keyword(s):  
Asymptotic Behaviour ◽  
Special Interest ◽  
Eigenvalue Problems ◽  
Asymptotic Integration ◽  
Linear Matrix ◽  
Linear Transformations ◽  
Matrix Differential ◽  
The Stability ◽  
Image Position ◽  
Index Problem

Consider the ordinary linear matrix differential systemψ(x) is a scalar mapping, X and A(x) are n by n matrices. Both belong to C1([a,∞)) for some integer l. The stability and asymptotic behaviour of its solutions have been subject to much investigation. See Bellman [2], Levinson [24], Hartman and Wintner [20], Devinatz [9], Fedoryuk [11], Harris and Lutz [16,17,18] and Cassell [30]. The special interest in eigenvalue problems and in the deficiency index problem stimulated a continued interest in asymptotic integration. See e.g. Naimark [36], Eastham and Grundniewicz [10] and [8,9]. Harris and Lutz [16,17,18] succeeded in explaining how to derive many known theorems in asymptotic integration by repeatedly using certain “(1 + Q)” linear transformations.

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