Correction of finite element eigenvalues for problems with natural or periodic boundary conditions

1988 ◽  
Vol 28 (2) ◽  
pp. 254-269 ◽  
Author(s):  
Alan L. Andrew
Keyword(s):  
Finite Element ◽  
Boundary Conditions ◽  
Periodic Boundary ◽  
Periodic Boundary Conditions

A numerical homogenization technique for unidirectional composites using polygonal generalized finite elements

2021 ◽  
pp. 146442072110463
Author(s):  
Murilo Sartorato
Keyword(s):  
Finite Element ◽  
Finite Elements ◽  
Boundary Conditions ◽  
Computational Efficiency ◽  
Periodic Boundary ◽  
Deformation Gradient ◽  
Reaction Force ◽  
Periodic Boundary Conditions ◽  
Unit Cell ◽  
Unit Cells

The present study proposes a computational methodology to obtain the homogenized effective elastic properties of unidirectional fibrous composite materials by using the generalized finite-element method and penalization techniques to impose periodic boundary conditions on non-uniform polygonal unit cells. Each unit cell is described by a single polygonal finite element using Wachspress functions as base shape functions and different families of enrichment functions to account for the internal fiber influence on stresses and strains fields. The periodic boundary conditions are imposed using reflection laws between two parallel opposing faces using a Lagrange multiplier approach; this reflection law creates a distributed reaction force over the edges of the [Formula: see text]-gon from the direct application of a given deformation gradient, which simulates different macroscopic load cases on the macroscopic body the unit cell is part of. The methodology is validated through a comparison with results for similar unit cells found in the literature and its computational efficiency is compared to simple cases solved using a classic finite-element approach. This methodology showed computational advantages over the classic finite elements in both computational efficiency and total number of degrees of freedom for convergence and flexibility on the shape of the unit cell used. Finally, the methodology provides an efficient way to introduce non-circular fiber shapes and voids.

The Energetics of Dislocation-Obstacle Interactions by 3-D Quasicontinuum Simulations

1999 ◽  
Vol 578 ◽  
Author(s):  
Kedar Hardikar ◽  
R. Phillips
Keyword(s):  
Finite Element ◽  
Boundary Conditions ◽  
Point Defects ◽  
Periodic Boundary ◽  
Boundary Effect ◽  
Periodic Boundary Conditions ◽  
Boundary Effects ◽  
Quasicontinuum Method ◽  
Finite Element Calculations ◽  
Infinite Extent

AbstractThe goal of this work is to study the interaction of dislocations with local obstacles to glide such as point defects, precipitates and other dislocations. The quasicontinuum method is used as the basis of this study. It is demonstrated that two types of boundary effects are of concern in the calculation of hardening parameters using finite sized simulation cells. A recently developed technique to incorporate periodic boundary conditions in the quasicontinuum method is used to eliminate surface effects which were present in earlier implementations and to simulate a dislocation of infinite extent interacting with an array of obstacles. The second type of boundary effect is due to the boundary conditions on the lateral boundaries. A method based on finite element calculations is proposed for quantifying the effect of lateral boundaries in these simulations. Preliminary results for the validation of the method are presented as well as a simulation of the interaction between a conventional edge dislocation in Al with an array of clusters of Ni atoms.

Micromechanical Modeling of Composites: A Local Approach for Imposing Periodic Boundary Conditions

2006 ◽  
Author(s):  
J. M. Tyrus ◽  
M. Gosz ◽  
E. DeSantiago
Keyword(s):  
Finite Element ◽  
Composite Materials ◽  
Boundary Conditions ◽  
Periodic Boundary ◽  
Periodic Boundary Conditions ◽  
Constitutive Law ◽  
Micromechanical Modeling ◽  
Coupling Scheme ◽  
Model Composite ◽  
Computational Speed

Recent advances in computational speed have resulted in the ability to model composite materials using larger representative volume elements (RVEs) with greater numbers of inclusions than have been previously studied. It is often necessary to assume periodicity for the effective evaluation of material properties, failure analysis, or constitutive law development for composite materials. Imposing periodic boundary conditions on very large RVEs can mean enforcing thousands of constraint equations. In addition, most commercial finite element codes incorporate these constraints on a global level, thereby drastically reducing computational speed. The present study investigates a method that uses a local implementation of the constraints that does not adversely affect the computational speed. As a step toward a three-dimensional formulation, the present study utilizes a two-dimensional triangular RVE of a periodically-spaced regular hexagonal array of composite material containing fibers of equal radii. In the present study, the finite element method is employed to obtain the response of the RVE. To impose the boundary conditions along the edges, this study utilizes a cubic interpolant to model the displacement field along the matrix edges and a linear interpolant to model the field along the fiber edges. The method eliminates the need for the conventional node-coupling scheme for imposing periodic boundary conditions, consequently reducing the number of unknowns to the interior degrees of freedom of the RVE along with a finite number of global parameters. The method results in a valuable computational savings that greatly simplifies the pre-processing stage of the analysis.

Sign in / Sign up

Export Citation Format

Share Document