An Effective Improved Algorithm for Finite Particle Method

2016 ◽  
Vol 13 (04) ◽  
pp. 1641009 ◽  
Author(s):  
Yang Yang ◽  
Fei Xu ◽  
Meng Zhang ◽  
Lu Wang
Keyword(s):  
Computing Time ◽  
Matrix Decomposition ◽  
Particle Method ◽  
Coefficient Matrix ◽  
Computational Accuracy ◽  
Sph Method ◽  
Modified Method ◽  
Transient Problems ◽  
Basic Equations ◽  
Finite Particle

The low accuracy near the boundary or the interface in SPH method has been paid extensive attention. The Finite Particle Method (FPM) is a significant improvement to the traditional SPH method, which can greatly improve the computational accuracy for boundary particles. However, there are still some inherent defects for FPM, such as the long computing time and the potential numerical instability. By conducting matrix decomposition on the basic equations of FPM, an improved FPM method (IFPM) is proposed, which can not only maintain the high accuracy of FPM for boundary particles, but also keep the invertibility of the coefficient matrix in FPM. The numerical results show that the IFPM is really an effective improvement to traditional FPM, which could greatly reduce the computing time. Finally, the modified method is applied to two transient problems.

scholarly journals Study on Particle Inconsistency Problem in Smoothed Particle Hydrodynamics

2011 ◽  
Vol 94-96 ◽  
pp. 1638-1641 ◽  
Author(s):  
Gui Ming Rong ◽  
Hiroyuki Kisu
Keyword(s):  
Smoothed Particle Hydrodynamics ◽  
Particle Method ◽  
Second Derivative ◽  
Sph Method ◽  
New Approach ◽  
First Derivative ◽  
Numerical Studies ◽  
Particle Hydrodynamics ◽  
Smoothed Particle ◽  
Finite Particle

In the smoothed particle hydrodynamics (SPH) method, the particle inconsistency problem significantly influences the calculation accuracy. In the present study, we investigate primarily the influence of the particle inconsistency on the first derivative of field functions and discuss the behavior of several methods of addressing this problem. In addition, we propose a new approach by which to compensate for this problem, especially for functions having a non-zero second derivative, that is less computational demanding, as compared to the finite particle method (FPM). A series of numerical studies have been carried out to verify the performance of the new approach.

A Kind of Preconditioned IGMRES(m) Algorithm

2012 ◽  
Vol 482-484 ◽  
pp. 413-416
Author(s):  
Chun Xiao Yu
Keyword(s):  
Krylov Subspace ◽  
Matrix Decomposition ◽  
Coefficient Matrix ◽  
Algebraic Equations ◽  
Residual Method ◽  
Minimal Residual Method ◽  
Algorithm Convergence ◽  
Minimal Residual

Fundamental theories are studied for an Incomplete Generalized Minimal Residual Method(IGMRES(m)) in Krylov subspace. An algebraic equations generated from the IGMRES(m) algorithm is presented. The relationships are deeply researched for the algorithm convergence and the coefficient matrix of the equations. A kind of preconditioned method is proposed to improve the convergence of the IGMRES(m) algorithm. It is proved that the best convergence can be obtained through appropriate matrix decomposition.

scholarly journals MN-PGSOR Method for Solving Nonlinear Systems with Block Two-by-Two Complex Symmetric Jacobian Matrices

2021 ◽  
Vol 2021 ◽  
pp. 1-18
Author(s):  
Yu-Ye Feng ◽  
Qing-Biao Wu
Keyword(s):  
Nonlinear Systems ◽  
Iteration Method ◽  
Computing Time ◽  
Coefficient Matrix ◽  
New Method ◽  
Jacobian Matrices ◽  
Modified Newton Method ◽  
Preconditioned Matrix ◽  
Outer Iteration ◽  
Inner Iteration

For solving the large sparse linear systems with 2 × 2 block structure, the generalized successive overrelaxation (GSOR) iteration method is an efficient iteration method. Based on the GSOR method, the PGSOR method introduces a preconditioned matrix with a new parameter for the coefficient matrix which can enhance the efficiency. To solve the nonlinear systems in which the Jacobian matrices are complex and symmetric with the block two-by-two form, we try to use the PGSOR method as an inner iteration, with the help of the modified Newton method as an efficient outer iteration method. This new method is called the modified Newton-PGSOR (MN-PGSOR) method. Local convergence properties of the MN-PGSOR are analyzed under the Hölder condition. Finally, we give the comparison of our new method with some previous methods in the numerical results. The MN-PGSOR method is superior in both iteration steps and computing time.

The Method of Fundamental Solutions for Solving Exterior Axisymmetric Helmholtz Problems with High Wave-Number

2013 ◽  
Vol 5 (04) ◽  
pp. 477-493 ◽  
Author(s):  
Wen Chen ◽  
Ji Lin ◽  
C.S. Chen
Keyword(s):  
Large Scale ◽  
Three Dimensional ◽  
Matrix Decomposition ◽  
Coefficient Matrix ◽  
Fundamental Solutions ◽  
Method Of Fundamental Solutions ◽  
Wave Number ◽  
High Wave Number ◽  
Memory Space ◽  
High Wave

AbstractIn this paper, we investigate the method of fundamental solutions (MFS) for solving exterior Helmholtz problems with high wave-number in axisymmetric domains. Since the coefficient matrix in the linear system resulting from the MFS approximation has a block circulant structure, it can be solved by the matrix decomposition algorithm and fast Fourier transform for the fast computation of large-scale problems and meanwhile saving computer memory space. Several numerical examples are provided to demonstrate its applicability and efficacy in two and three dimensional domains.

Application of an Implicit Relaxation Method Solving the Euler Equations for Time-Accurate Unsteady Problems

1990 ◽  
Vol 112 (4) ◽  
pp. 510-520 ◽  
Author(s):  
A. Brenneis ◽  
A. Eberle
Keyword(s):  
Euler Equations ◽  
Computing Time ◽  
Three Dimensional ◽  
Numerical Procedure ◽  
Relaxation Method ◽  
Coefficient Matrix ◽  
Viscous Effects ◽  
Left Hand ◽  
Implicit Equations ◽  
The Right

A numerical procedure is presented for computing time-accurate solutions of flows about two and three-dimensional configurations using the Euler equations in conservative form. A nonlinear Newton method is applied to solve the unfactored implicit equations. Relaxation is performed with a point Gauss-Seidel algorithm ensuring a high degree of vectorization by employing the so-called checkerboard scheme. The fundamental feature of the Euler solver is a characteristic variable splitting scheme (Godunov-type averaging procedure, linear locally one-dimensional Riemann solver) based on an eigenvalue analysis for the calculation of the fluxes. The true Jacobians of the fluxes on the right-hand side are used on the left-hand side of the first order in time-discretized Euler equations. A simple matrix conditioning needing only few operations is employed to evade singular behavior of the coefficient matrix. Numerical results are presented for transonic flows about harmonically pitching airfoils and wings. Comparisons with experiments show good agreement except in regions where viscous effects are evident.

Tensegrity Form-Finding Using Finite Particle Method

2011 ◽  
Vol 201-203 ◽  
pp. 1166-1169
Author(s):  
Xian Xu ◽  
Ying Yu ◽  
Yao Zhi Luo ◽  
Yan Bin Shen
Keyword(s):  
Large Scale ◽  
Particle Method ◽  
Relaxation Method ◽  
Dynamic Relaxation ◽  
Form Finding ◽  
Mathematical Functions ◽  
Tensegrity Systems ◽  
Effective Form ◽  
Dynamic Relaxation Method ◽  
Finite Particle

In this paper, a new method, so-called finite particle method, for form-finding of tensegrity systems is proposed. It models the analyzed domain by finite particles instead of mathematical functions and continuous bodies in traditional mechanics. Examples including simple regular tensegrity systems and complex irregular tensegrity systems are carried out to verify the feasibility and effectiveness of the method. Also a comparison between the proposed method and the dynamic relaxation method is conducted. It proves that the proposed method is much more effective than the dynamic relaxation method in the form-finding of large-scale tensegrity systems. Hence, the proposed method has a great potential of developing into a general and effective form-finding method for irregular and large-scale tensegrity systems.

An Improved Specified Finite Particle Method and Its Application to Transient Heat Conduction

2016 ◽  
Vol 14 (05) ◽  
pp. 1750050 ◽  
Author(s):  
Lu Wang ◽  
Fei Xu ◽  
Yang Yang
Keyword(s):  
Heat Conduction ◽  
Matrix Theory ◽  
Taylor Series Expansion ◽  
Particle Method ◽  
Transient Heat Conduction ◽  
Boundary Region ◽  
The Matrix ◽  
Particle Hydrodynamics ◽  
Smoothed Particle ◽  
Finite Particle

Compared with the traditional Smoothed Particle Hydrodynamics (SPH), Finite Particle Method (FPM) has higher accuracy for boundary region. However, there are still two inherent defects which are the time consuming and the numerical instability in FPM. In this paper, a high-order algorithm based on the Taylor series expansion and the matrix theory is proposed and the corresponding particles selected mode is discussed. It is validated that the algorithm has higher-order accuracy than the previous low-order improvement algorithm for FPM. Further, transient heat conduction examples have been discussed to verify the feasibility and effectiveness of the new algorithm.

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