scholarly journals Generalized Neumann Expansion and Its Application in Stochastic Finite Element Methods

2013 ◽  
Vol 2013 ◽  
pp. 1-13 ◽  
Author(s):  
Xiangyu Wang ◽  
Song Cen ◽  
Chenfeng Li
Keyword(s):  
Finite Element ◽  
Perturbation Method ◽  
High Efficiency ◽  
Expansion Method ◽  
Error Estimator ◽  
Algebraic Equations ◽  
Stochastic Finite Element ◽  
Stochastic Field ◽  
Element Analysis ◽  
Linear Algebraic Equations

An acceleration technique, termed generalized Neumann expansion (GNE), is presented for evaluating the responses of uncertain systems. The GNE method, which solves stochastic linear algebraic equations arising in stochastic finite element analysis, is easy to implement and is of high efficiency. The convergence condition of the new method is studied, and a rigorous error estimator is proposed to evaluate the upper bound of the relative error of a given GNE solution. It is found that the third-order GNE solution is sufficient to achieve a good accuracy even when the variation of the source stochastic field is relatively high. The relationship between the GNE method, the perturbation method, and the standard Neumann expansion method is also discussed. Based on the links between these three methods, quantitative error estimations for the perturbation method and the standard Neumann method are obtained for the first time in the probability context.

Stochastic Finite Element Analysis for High Speed Rotors

1993 ◽  
Vol 115 (1) ◽  
pp. 59-64 ◽  
Author(s):  
T. S. Sankar ◽  
S. A. Ramu ◽  
R. Ganesan
Keyword(s):  
Finite Element ◽  
High Speed ◽  
Mass Density ◽  
Adjoint System ◽  
Stochastic Finite Element ◽  
Stochastic Field ◽  
Rotating Shaft ◽  
Stochastic Variation ◽  
Element Analysis ◽  
Stochastic Finite Element Analysis

The general problem of the dynamic response of highspeed rotors is considered in which certain system parameters may have a spatial stochastic variation. In particular the elastic modulus and mass density of a rotating shaft are described through one dimensional stochastic field functions so that the imperfections in manufacture and measurement can be accounted for. The stochastic finite element method is developed so that the variability of the response of the rotor can be interpreted in terms of the variation of the material property. As an illustration the whirl speed analysis is performed to determine the stochastics of whirl speeds and modes through the solution of a random eigenvalue problem associated with a non self-adjoint system. Numerical results are also presented.

scholarly journals Stochastic Finite Element Technique for Stochastic One-Dimension Time-Dependent Differential Equations with Random Coefficients

2007 ◽  
Vol 2007 ◽  
pp. 1-16 ◽  
Author(s):  
M. M. Saleh ◽  
I. L. El-Kalla ◽  
M. M. Ehab
Keyword(s):  
Finite Element ◽  
Differential Equations ◽  
Mass Density ◽  
Solution Process ◽  
Random Coefficients ◽  
Time Dependent ◽  
One Dimension ◽  
Algebraic Equations ◽  
Stochastic Finite Element ◽  
Linear Algebraic Equations

The stochastic finite element method (SFEM) is employed for solving stochastic one-dimension time-dependent differential equations with random coefficients. SFEM is used to have a fixed form of linear algebraic equations for polynomial chaos coefficients of the solution process. Four fixed forms are obtained in the cases of stochastic heat equation with stochastic heat capacity or heat conductivity coefficients and stochastic wave equation with stochastic mass density or elastic modulus coefficients. The relation between the exact deterministic solution and the mean of solution process is numerically studied.

scholarly journals Stability of orthotropic plates

2020 ◽  
Vol 166 ◽  
pp. 06004
Author(s):  
Mykola Surianinov ◽  
Dina Lazarieva ◽  
Iryna Kurhan
Keyword(s):  
Differential Equation ◽  
Finite Element ◽  
Boundary Conditions ◽  
Critical Loads ◽  
Orthotropic Plate ◽  
Algebraic Equations ◽  
Stability Equation ◽  
Element Analysis ◽  
Linear Algebraic Equations ◽  
The Stability

The solution to the problem of the stability of a rectangular orthotropic plate is described by the numerical-analytical method of boundary elements. As is known, the basis of this method is the analytical construction of the fundamental system of solutions and Green’s functions for the differential equation (or their system) for the problem under consideration. To account for certain boundary conditions, or contact conditions between the individual elements of the system, a small system of linear algebraic equations is compiled, which is then solved numerically. It is shown that four combinations of the roots of the characteristic equation corresponding to the differential equation of the problem are possible, which leads to the need to determine sixty-four analytical expressions of fundamental functions. The matrix of fundamental functions, which is the basis of the transcendental stability equation, is very sparse, which significantly improves the stability of numerical operations and ensures high accuracy of the results. An analysis of the numerical results obtained by the author’s method shows very good convergence with the results of finite element analysis. For both variants of the boundary conditions, the discrepancy for the corresponding critical loads is almost the same, and increases slightly with increasing critical load. Moreover, this discrepancy does not exceed one percent. It is noted that under both variants of the boundary conditions, the critical loads calculated by the boundary element method are less than in the finite element calculations. The obtained transcendental stability equation allows to determine critical forces both by the static method and by the dynamic one. From this equation it is possible to obtain a spectrum of critical forces for a fixed number of half-waves in the direction of one of the coordinate axes. The proposed approach allows us to obtain a solution to the stability problem of an orthotropic plate under any homogeneous and inhomogeneous boundary conditions.

scholarly journals IMPLEMENTATION OF A FINITE ELEMENT CLASS LIBRARY USING GENERALIZED PROGRAMMING

2021 ◽  
pp. 164-173
Author(s):  
S. V. Choporov ◽  
M. S. Ihnatchenko ◽  
O. V. Kudin ◽  
A. G. Kryvokhata ◽  
S. I. Homeniuk
Keyword(s):  
Finite Element Analysis ◽  
Finite Element ◽  
General Purpose ◽  
Open Architecture ◽  
Algebraic Equations ◽  
Complex Objects ◽  
Element Analysis ◽  
Class Library ◽  
Linear Algebraic Equations ◽  
Numerical Finite Element

Context. For computer modeling of complex objects and phenomena of various nature, in practice, the numerical finite element method is often used. Its software implementation (especially for the study of new classes of problems) is a rather laborious process. The high cost of software development makes the development of new approaches to improving the efficiency of programming and maintenance (including the addition of new functions) urgent. Objective. The aim of the work is to create a new effective architecture of programs for finite element analysis of problems in mathematical physics, which makes it easy to expand their functionality to solve new classes of problems. Method. A method for developing programs for finite element analysis using generalized programming is proposed, which makes it possible to significantly simplify the architecture of the software and make it more convenient for maintenance and modification by separating algorithms and data structures. A new architecture of classes that implement finite element calculation is proposed, which makes it possible to easily expand the functionality of programs by adding new types of finite elements, methods for solving systems of linear algebraic equations, parallel computations, etc. Results. The proposed approach was implemented in software as a class library in C ++. A number of computational experiments have been carried out, which have confirmed its efficiency in solving practical problems. Conclusions. The developed approach can be used both to create general-purpose finite element analysis systems with an open architecture, and to implement specialized software packages focused on solving specific classes of problems (fracture mechanics, elastomers, contact interaction, etc.).

Finite‐element analysis of controlled‐source electromagnetic induction using Coulomb‐gauged potentials

Geophysics ◽  
2001 ◽  
Vol 66 (3) ◽  
pp. 786-799 ◽  
Author(s):  
Eugene A. Badea ◽  
Mark E. Everett ◽  
Gregory A. Newman ◽  
Oszkar Biro
Keyword(s):  
Finite Element ◽  
Heterogeneous Media ◽  
Petroleum Exploration ◽  
Algebraic Equations ◽  
Weak Formulation ◽  
Element Analysis ◽  
Controlled Source ◽  
Linear Algebraic Equations ◽  
Em Induction ◽  
Log Interpretation

A 3-D finite‐element solution has been used to solve controlled‐source electromagnetic (EM) induction problems in heterogeneous electrically conducting media. The solution is based on a weak formulation of the governing Maxwell equations using Coulomb‐gauged EM potentials. The resulting sparse system of linear algebraic equations is solved efficiently using the quasi‐minimal residual method with simple Jacobi scaling as a preconditioner. The main aspects of this work include the implementation of a 3-D cylindrical mesh generator with high‐quality local mesh refinement and a formulation in terms of secondary EM potentials that eliminates singularities introduced by the source. These new aspects provide quantitative induction‐log interpretation for petroleum exploration applications. Examples are given for 1-D, 2-D, and 3-D problems, and favorable comparisons are presented against other, previously published multidimensional EM induction codes. The method is general and can also be adapted for controlled‐source EM modeling in mining, groundwater, and environmental geophysics in addition to fundamental studies of EM induction in heterogeneous media.

Finite Element Modeling for Steel Cord Analysis in Radial Tires

2013 ◽  
Vol 41 (1) ◽  
pp. 60-79 ◽  
Author(s):  
Wei Yintao ◽  
Luo Yiwen ◽  
Miao Yiming ◽  
Chai Delong ◽  
Feng Xijin
Keyword(s):  
Finite Element ◽  
High Efficiency ◽  
Force Measurement ◽  
Three Dimensional ◽  
Local Stress ◽  
Tension Force ◽  
Center Line ◽  
Element Analysis ◽  
Design Variables ◽  
Steel Cord

ABSTRACT: This article focuses on steel cord deformation and force investigation within heavy-duty radial tires. Typical bending deformation and tension force distributions of steel reinforcement within a truck bus radial (TBR) tire have been obtained, and they provide useful input for the local scale modeling of the steel cord. The three-dimensional carpet plots of the cord force distribution within a TBR tire are presented. The carcass-bending curvature is derived from the deformation of the carcass center line. A high-efficiency modeling approach for layered multistrand cord structures has been developed that uses cord design variables such as lay angle, lay length, and radius of the strand center line as input. Several types of steel cord have been modeled using the developed method as an example. The pure tension for two cords and the combined tension bending under various loading conditions relevant to tire deformation have been simulated by a finite element analysis (FEA). Good agreement has been found between experimental and FEA-determined tension force-displacement curves, and the characteristic structural and plastic deformation phases have been revealed by the FE simulation. Furthermore, some interesting local stress and deformation patterns under combined tension and bending are found that have not been previously reported. In addition, an experimental cord force measurement approach is included in this article.

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