Finite Element Error Localization Using the Error Matrix Method

1997 ◽  
Author(s):  
Michael Papadopoulos ◽  
Ephrahim Garcia
Keyword(s):  
Finite Element ◽  
Matrix Method ◽  
Degrees Of Freedom ◽  
Higher Order ◽  
Bernoulli Beam ◽  
Error Matrix ◽  
Error Localization ◽  
Higher Order Terms ◽  
Euler Bernoulli Beam ◽  
Element Error

Abstract A higher order version of the Error Matrix Method is proposed to increase the accuracy in the finite element error localization. The method retains a user specified number of terms from the appropriate binomial expansion. Jacobi’s iterative method is then proposed to solve the set of nonlinear equations. It is hypothesized that keeping the higher order terms will improve the error identification for the same number of coordinate degrees-of-freedom and modes. The method is implemented on a nine degree-of-freedom and an Euler-Bernoulli beam numerical examples. It is shown that while there needs to be a large number of measured coordinates and modes, the magnitude of the errors are more accurately identified.

Curvature-Based Finite Element Method for Euler-Bernoulli Beams

2007 ◽  
Author(s):  
Y. L. Kuo ◽  
W. L. Cleghorn
Keyword(s):  
Finite Element Method ◽  
Finite Element ◽  
Degrees Of Freedom ◽  
Transverse Displacement ◽  
Bernoulli Beam ◽  
Numerical Examples ◽  
Displacement Based ◽  
Curvature Distribution ◽  
Euler Bernoulli Beam ◽  
Element Method

This paper presents a new method called the curvature-based finite element method to solve Euler-Bernoulli beam problems. An approximated curvature distribution is selected first, and then the approximated transverse displacement is determined by double integrations. Four numerical examples demonstrate the validity of the method, and the results show that the errors are smaller than those generated by a conventional method, the displacement-based finite element method, for comparison based on the same number of degrees of freedom.

Damping Improvement in Layered and Jointed Beams by Finite Element Analysis

2012 ◽  
Vol 505 ◽  
pp. 501-505 ◽  
Author(s):  
D.N. Thatoi ◽  
R.C. Mohanty ◽  
A.K. Acharya ◽  
B.K. Nanda
Keyword(s):  
Finite Element ◽  
Degrees Of Freedom ◽  
Beam Theory ◽  
Element Model ◽  
Damping Capacity ◽  
Bernoulli Beam ◽  
Element Analysis ◽  
Linear Elastic ◽  
Mass Matrices ◽  
Euler Bernoulli Beam

Damping in built-up structures is produced by the energy dissipation due to micro-slip along the frictional interfaces. A finite element model of the linear elastic system has been formulated using the Euler-Bernoulli beam theory to investigate the damping phenomena in riveted connections. The discrete element system having two degrees of freedom per node representing v and has been used for the analysis. The generalized stiffness and mass matrices for this element has been derived. Extensive experiments have been conducted for the validation of the analysis. From this study, it is established that the damping capacity increases and the natural frequency decreases due to the joint effects.

scholarly journals Higher Order Multiscale Finite Element Method for Heat Transfer Modeling

Materials ◽  
2021 ◽  
Vol 14 (14) ◽  
pp. 3827
Author(s):  
Marek Klimczak ◽  
Witold Cecot
Keyword(s):  
Heat Transfer ◽  
Finite Element Method ◽  
Finite Element ◽  
Degrees Of Freedom ◽  
Higher Order ◽  
Shape Functions ◽  
Multiscale Finite Element Method ◽  
Multiscale Finite Element ◽  
Steady State Heat ◽  
Steady State Heat Transfer

In this paper, we present a new approach to model the steady-state heat transfer in heterogeneous materials. The multiscale finite element method (MsFEM) is improved and used to solve this problem. MsFEM is a fast and flexible method for upscaling. Its numerical efficiency is based on the natural parallelization of the main computations and their further simplifications due to the numerical nature of the problem. The approach does not require the distinct separation of scales, which makes its applicability to the numerical modeling of the composites very broad. Our novelty relies on modifications to the standard higher-order shape functions, which are then applied to the steady-state heat transfer problem. To the best of our knowledge, MsFEM (based on the special shape function assessment) has not been previously used for an approximation order higher than p = 2, with the hierarchical shape functions applied and non-periodic domains, in this problem. Some numerical results are presented and compared with the standard direct finite-element solutions. The first test shows the performance of higher-order MsFEM for the asphalt concrete sample which is subject to heating. The second test is the challenging problem of metal foam analysis. The thermal conductivity of air and aluminum differ by several orders of magnitude, which is typically very difficult for the upscaling methods. A very good agreement between our upscaled and reference results was observed, together with a significant reduction in the number of degrees of freedom. The error analysis and the p-convergence of the method are also presented. The latter is studied in terms of both the number of degrees of freedom and the computational time.

Asymmetric Bifurcation of Initially Curved Nanobeam

2015 ◽  
Vol 82 (9) ◽  
Author(s):  
X. Chen ◽  
S. A. Meguid
Keyword(s):  
Symmetry Breaking ◽  
Degrees Of Freedom ◽  
Surface Elasticity ◽  
Beam Theory ◽  
Surface Effects ◽  
Beam Model ◽  
Bernoulli Beam ◽  
Reduced Order ◽  
Euler Bernoulli Beam ◽  
Bifurcation Behavior

In this paper, we investigate the asymmetric bifurcation behavior of an initially curved nanobeam accounting for Lorentz and electrostatic forces. The beam model was developed in the framework of Euler–Bernoulli beam theory, and the surface effects at the nanoscale were taken into account in the model by including the surface elasticity and the residual surface tension. Based on the Galerkin decomposition method, the model was simplified as two degrees of freedom reduced order model, from which the symmetry breaking criterion was derived. The results of our work reveal the significant surface effects on the symmetry breaking criterion for the considered nanobeam.

Matrix Method for Buckling Analysis of Frames Based on Hencky Bar-Chain Model

2019 ◽  
Vol 19 (08) ◽  
pp. 1950093 ◽  
Author(s):  
W. H. Pan ◽  
C. M. Wang ◽  
H. Zhang
Keyword(s):  
Matrix Method ◽  
Structural Changes ◽  
Beam Theory ◽  
Buckling Analysis ◽  
Chain Model ◽  
Bernoulli Beam ◽  
Various Boundary Conditions ◽  
End Restraint ◽  
Euler Bernoulli Beam ◽  
General Frames

Presented herein is a matrix method for buckling analysis of general frames based on the Hencky bar-chain model comprising of rigid segments connected by hinges with elastic rotational springs. Unlike the conventional matrix method of structural analysis based on the Euler–Bernoulli beam theory, the Hencky bar-chain model (HBM) matrix method allows one to readily handle the localized changes in end restraint conditions or localized structural changes (such as local damage or local stiffening) by simply tweaking the spring stiffnesses. The developed HBM matrix method was applied to solve some illustrative example problems to demonstrate its versatility in solving the buckling problem of beams and frames with various boundary conditions and local changes. It is hoped that this easy-to-code HBM matrix method will be useful to engineers in solving frame buckling problems.

scholarly journals Higher order multipoint flux mixed finite element methods on quadrilaterals and hexahedra

2019 ◽  
Vol 29 (06) ◽  
pp. 1037-1077 ◽  
Author(s):  
Ilona Ambartsumyan ◽  
Eldar Khattatov ◽  
Jeonghun J. Lee ◽  
Ivan Yotov
Keyword(s):  
Finite Element ◽  
Degrees Of Freedom ◽  
Mixed Finite Element ◽  
Mixed Finite Elements ◽  
Higher Order ◽  
Local Velocity ◽  
New Family ◽  
Gauss Points ◽  
Theoretical Results ◽  
Optimal Formula

We develop higher order multipoint flux mixed finite element (MFMFE) methods for solving elliptic problems on quadrilateral and hexahedral grids that reduce to cell-based pressure systems. The methods are based on a new family of mixed finite elements, which are enhanced Raviart–Thomas spaces with bubbles that are curls of specially chosen polynomials. The velocity degrees of freedom of the new spaces can be associated with the points of tensor-product Gauss–Lobatto quadrature rules, which allows for local velocity elimination and leads to a symmetric and positive definite cell-based system for the pressures. We prove optimal [Formula: see text]th order convergence for the velocity and pressure in their natural norms, as well as [Formula: see text]st order superconvergence for the pressure at the Gauss points. Moreover, local postprocessing gives a pressure that is superconvergent of order [Formula: see text] in the full [Formula: see text]-norm. Numerical results illustrating the validity of our theoretical results are included.

Developing a beam formulation for semi-crystalline two-way shape memory polymers

2020 ◽  
Vol 31 (12) ◽  
pp. 1465-1476
Author(s):  
Mohammad-Ali Maleki-Bigdeli ◽  
Majid Baniassadi ◽  
Kui Wang ◽  
Mostafa Baghani
Keyword(s):  
Finite Element ◽  
Shape Memory ◽  
Shape Memory Polymer ◽  
Beam Theory ◽  
Shape Memory Polymers ◽  
Bernoulli Beam ◽  
Von Karman ◽  
Von Kármán ◽  
Euler Bernoulli Beam ◽  
Beam Theories

In this research, the bending of a two-way shape memory polymer beam is examined implementing a one-dimensional phenomenological macroscopic constitutive model into Euler–Bernoulli and von-Karman beam theories. Since bending loading is a fundamental problem in engineering applications, a combination of bending problem and two-way shape memory effect capable of switching between two temporary shapes can be used in different applications, for example, thermally activated sensors and actuators. Shape memory polymers as a branch of soft materials can undergo large deformation. Hence, Euler–Bernoulli beam theory does not apply to the bending of a shape memory polymer beam where moderate rotations may occur. To overcome this limitation, von-Karman beam theory accounting for the mid-plane stretching as well as moderate rotations can be employed. To investigate the difference between the two beam theories, the deflection and rotating angles of a shape memory polymer cantilever beam are analyzed under small and moderate deflections and rotations. A semi-analytical approach is used to inspect Euler–Bernoulli beam theory, while finite-element method is employed to study von-Karman beam theory. In the following, a smart structure is analyzed using a prepared user-defined subroutine, VUMAT, in finite-element package, ABAQUS/EXPLICIT. Utilizing generated user-defined subroutine, smart structures composed of shape memory polymer material can be analyzed under complex loading circumstances through the two-way shape memory effect.

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