scholarly journals Application Of The Interlaced Sweep Method For The Solution Of Problems In Field Theory

2015 ◽  
Vol 3 ◽  
pp. 170
Author(s):  
Aloizs Ratnieks ◽  
Marina Uhanova
Keyword(s):  
Field Theory ◽  
Finite Element Method ◽  
Finite Element ◽  
Direct Methods ◽  
System Of Equations ◽  
Algebraic Equations ◽  
The Finite Element Method ◽  
Sweep Method ◽  
Linear Algebraic Equations ◽  
Separate Block

<p class="R-AbstractKeywords"><span lang="EN-US">For solution of problems in field theory the method of sweep is very popular. The authors suggest a very effective method of interlaced sweep. The essence of the interlaced sweep method lies in the fact that matrix of the linear algebraic equations system is broken into parts and the solution of separate blocks is sought by direct methods. Usually for each line of the grid a separate block is created. The system of block equations has a tridiagonal matrix where only elements of the main diagonal and two neighboring diagonals are different from zero. The system of equations with such a matrix is easily solved by the method of elimination of unknowns.</span></p><p class="R-AbstractKeywords"><span lang="EN-US">By solving the problems by the finite element method, the nodes of touching of neighboring elements can be placed on curved lines, and the sweep on these lines can be performed observing the principle of interlaced sweep. By following this method, the neighboring lines should not belong to the same half-step.</span></p>

Calculation of the Blade-to-Blade Compressible Flow Field in Turbo Impellers Using the Finite-Element Method

1977 ◽  
Vol 19 (3) ◽  
pp. 108-112 ◽  
Author(s):  
D. Adler ◽  
Y. Krimerman
Keyword(s):  
Finite Element Method ◽  
Finite Element ◽  
Variational Approach ◽  
Iterative Procedure ◽  
Coefficient Matrix ◽  
Algebraic Equations ◽  
The Finite Element Method ◽  
Variational Functional ◽  
Linear Algebraic Equations ◽  
Computer Storage

No variational principle can be found for Wu's blade-to-blade equation and therefore no appropriate variational functional associated with the problem can be derived. This difficulty is overcome by using a Poisson equation as the basis for an iterative procedure. Thus the method retains the advantage of the variational approach in which the coefficient matrix of the linear algebraic equations is always symmetric. The symmetry of the coefficient matrix allows reduction of computer storage.

Schwarz Type Solvers for -FEM Discretizations of Mixed Problems

2012 ◽  
Vol 12 (4) ◽  
pp. 369-390
Author(s):  
Sven Beuchler ◽  
Martin Purrucker
Keyword(s):  
Finite Element ◽  
Stokes Problem ◽  
Schur Complement ◽  
Algebraic Equations ◽  
The Finite Element Method ◽  
Contraction Ratio ◽  
Linear Algebraic Equations ◽  
Stokes Problems ◽  
Elasticity Problems ◽  
Element Pair

AbstractThis paper investigates the discretization of mixed variational formulation as, e.g., the Stokes problem by means of the hp-version of the finite element method. The system of linear algebraic equations is solved by the preconditioned Bramble-Pasciak conjugate gradient method. The development of an efficient preconditioner requires three ingredients, a preconditioner related to the components of the velocity modes, a preconditioner for the Schur complement related to the components of the pressure modes and a discrezation by a stable finite element pair which satisfies the discrete inf-sup-condition. The last condition is also important in order to obtain a stable discretization scheme. The preconditioner for the velocity modes is adapted from fast $hp$-FEM preconditioners for the potential equation. Moreover, we will prove that the preconditioner for the Schur complement can be chosen as a diagonal matrix if the pressure is discretized by discontinuous finite elements. We will prove that the system of linear algebraic equations can be solved in almost optimal complexity. This yields quasioptimal hp-FEM solvers for the Stokes problems and the linear elasticity problems. The latter are robust with respect to the contraction ratio ν. The efficiency of the presented solver is shown in several numerical examples.

An Analysis of the Large Deflections of Beams using the Rayleigh-Ritz Finite Element Method

1968 ◽  
Vol 19 (4) ◽  
pp. 357-367 ◽  
Author(s):  
A. C. Walker ◽  
D. G. Hall
Keyword(s):  
Finite Element Method ◽  
Finite Element ◽  
Numerical Method ◽  
Large Deflection ◽  
Energy Functional ◽  
Algebraic Equations ◽  
Linear Algebraic Equations ◽  
Non Linear ◽  
Comparison Of The Results ◽  
Element Method

SummaryThe Rayleigh-Ritz finite element method is used to obtain an approximate solution of the exact non-linear energy functional describing the large deflection bending behaviour of a simply-supported inextensible uniform beam subjected to point loads. The solution of the non-linear algebraic equations resulting from the use of this method is effected, using three different techniques, and comparisons are made regarding the accuracy and computing effort involved in each. A description is given of an experimental investigation of the problem and comparison of the results with those of the numerical method, and of the available exact continuum analyses, indicates that the numerical method provides satisfactory predictions for the non-linear beam behaviour for deflections up to one quarter of the beam’s length.

scholarly journals Construction of an Effective Preconditioner for the Even-odd Splitting of Cubic Spline Wavelets

2021 ◽  
Vol 20 ◽  
pp. 717-728
Author(s):  
Boris M. Shumilov
Keyword(s):  
Band System ◽  
Cubic Spline ◽  
System Of Equations ◽  
Algebraic Equations ◽  
Diagonal Dominance ◽  
Sweep Method ◽  
Spline Wavelets ◽  
Linear Algebraic Equations ◽  
Initial Scale ◽  
Strict Diagonal Dominance

In this study, the method for decomposing splines of degree m and smoothness C^m-1 into a series of wavelets with zero moments is investigated. The system of linear algebraic equations connecting the coefficients of the spline expansion on the initial scale with the spline coefficients and wavelet coefficients on the embedded scale is obtained. The originality consists in the application of some preconditioner that reduces the system to a simpler band system of equations. Examples of applying the method to the cases of first-degree spline wavelets with two first zero moments and cubic spline wavelets with six first zero moments are presented. For the cubic case after splitting the system into even and odd rows, the resulting matrix acquires a seven-diagonals form with strict diagonal dominance, which makes it possible to apply an effective sweep method to its solution

Stabilized numerical method for solving the Oseen type problem with a singularity

2018 ◽  
Author(s):  
А.В. Рукавишников
Keyword(s):  
Finite Element Method ◽  
Finite Element ◽  
Krylov Subspace ◽  
Stokes Equations ◽  
Domain Decomposition Method ◽  
Algebraic Equations ◽  
Type Problem ◽  
Curvilinear Boundary ◽  
Linear Algebraic Equations ◽  
Element Method

На основе метода декомпозиции области построен стабилизационный неконформный метод конечных элементов для решения задачи типа Озеена. Для конвективно доминирующих течений с разрывным коэффициентом вязкости определена шкала оптимального выбора стабилизирующего параметра. Результаты численных экспериментов согласуются с теоретической оценкой сходимости. Purpose. To construct modified approximation approach using the finite element method and to perform numerical analysis for a two dimensional problem on the flow of a viscous inhomogeneous fluids — the Oseen type problem, that is obtained by sampling in time and linearizing the incompressible Navier—Stokes equations. To consider the convection dominated flow case. Methodology. Based on the domain decomposition method with a smooth curvilinear boundary between subdomains, a stabilization nonconformal finite element method is constructed that satisfies the inf-sup-stability condition. To solve the resulting system of linear algebraic equations, an iterative process is considered that uses the decomposition of the vector in the Krylov subspace with minimal inviscidity, with a block preconditioning of the matrix. Findings. The results of the numerical experiments demonstrate the robustness of the considered method for different (even small) discontinuous values of viscosity. The differences between finite element and exact solutions for the velocity field and pressure in the norms of the grid spaces decrease as

scholarly journals Improvement of the finite element method equations conditioning for the magnetic field-circuital problems

2017 ◽  
Vol 66 (2) ◽  
pp. 325-337
Author(s):  
Marek Gołębiowski
Keyword(s):  
Finite Element Method ◽  
Finite Element ◽  
Iterative Methods ◽  
Condition Number ◽  
Sparse Matrix ◽  
Solution Process ◽  
System Of Equations ◽  
System Matrix ◽  
The Finite Element Method ◽  
Element Method

AbstractThe presented systems with magnetically coupled windings are solved with the finite element method. If the issue of voltage supply is analyzed, a system of linear equations with a partially skew-symmetric sparse matrix is obtained. Iterative methods used to solve a system of equations are particularly effective for symmetric matrices. Resultant equations can be reduced to this symmetrical form by using the method known from the literature [1]. The ratio of the maximum to the minimum eigenvalue of the main matrix of this circuit, which is the condition number, is however very high. This means that the problem is ill-conditioned and leads to a very long iterative solution process. The method presented in the article allows for a direct solution of a system of equations on its part, corresponding to high eigenvalues of the system matrix. The remaining part of the system of equations is solved by iterative methods. This part has much better condition number, and therefore the computational process is fast. The proposed iterative process depends on multiplication of a sparse matrix by vectors. It is not necessary (and possible) to store the entire matrix. This is especially important for larger sizes of a matrix.

A New Mixed Finite Element–Differential Quadrature Formulation for Forced Vibration of Beams Carrying Moving Loads

2010 ◽  
Vol 78 (1) ◽  
Author(s):  
A. A. Jafari ◽  
S. A. Eftekhari
Keyword(s):  
Finite Element Method ◽  
Finite Element ◽  
Gaussian Elimination ◽  
Mixed Finite Element ◽  
Differential Quadrature ◽  
Time Dependent ◽  
Algebraic Equations ◽  
The Finite Element Method ◽  
Time Dependent Problems ◽  
Element Method

In this paper, a new version of mixed finite element–differential quadrature formulation is presented for solving time-dependent problems. The governing partial differential equation of motion of the structure is first reduced to a set of ordinary differential equations (ODEs) in time using the finite element method. The resulting system of ODEs is then satisfied at any discrete time point apart and changed to a set of algebraic equations by the application of differential quadrature method (DQM) for time derivative discretization. The resulting set of algebraic equations can be solved by either direct methods (such as the Gaussian elimination method) or iterative methods (such as the Gauss–Seidel method). The mixed formulation enjoys the strong geometry flexibility of the finite element method and the high accuracy, low computational efforts, and efficiency of the DQM. The application of the formulation is then shown by solving some moving load class of problems (i.e., moving force, moving mass, and moving oscillator problems). The stability property and computational efficiency of the scheme are also discussed in detail. Numerical results show that the proposed mixed methodology can be used as an efficient tool for handling the time-dependent problems.

Sign in / Sign up

Export Citation Format

Share Document