scholarly journals Weighted Finite-Element Method for Elasticity Problems with Singularity

2018 ◽  
Author(s):  
Viktor Anatolievich Rukavishnikov ◽  
Elena Ivanovna Rukavishnikova
Keyword(s):  
Finite Element Method ◽  
Finite Element ◽  
Elasticity Problems ◽  
Element Method

Finite Element Models for Flexible Cosserat Solids

2020 ◽  
Author(s):  
Olivier A. Bauchau ◽  
Minghe Shan
Keyword(s):  
Finite Element Method ◽  
Finite Element ◽  
Displacement Vector ◽  
Equations Of Motion ◽  
Three Dimensional ◽  
The Finite Element Method ◽  
Discrete Equations ◽  
Computation Efficiency ◽  
Elasticity Problems ◽  
Element Method

Abstract The application of the finite element method to the modeling of Cosserat solids is investigated in detail. In two- and three-dimensional elasticity problems, the nodal unknowns are the components of the displacement vector, which form a linear field. In contrast, when dealing with Cosserat solids, the nodal unknowns form the special Euclidean group SE(3), a nonlinear manifold. This observation has numerous implications on the implementation of the finite element method and raises numerous questions: (1) What is the most suitable representation of this nonlinear manifold? (2) How is it interpolated over one element? (3) How is the associated strain field interpolated? (4) What is the most efficient way to obtain the discrete equations of motion? All these questions are, of course intertwined. This paper shows that reliable schemes are available for the interpolation of the motion and curvature fields. The interpolated fields depend on relative nodal motions only, and hence, are both objective and tensorial. Because these schemes depend on relative nodal motions only, only local parameterization is required, thereby avoiding the occurrence of singularities. For Cosserat solids, it is preferable to perform the discretization operation first, followed by the variation operation. This approach leads to considerable computation efficiency and simplicity.

A Reduced Crouzeix–Raviart Immersed Finite Element Method for Elasticity Problems with Interfaces

2020 ◽  
Vol 20 (3) ◽  
pp. 501-516
Author(s):  
Gwanghyun Jo ◽  
Do Young Kwak
Keyword(s):  
Finite Element Method ◽  
Finite Element ◽  
Poisson Ratio ◽  
The Other ◽  
Optimal Order ◽  
Optimal Order Of Convergence ◽  
Immersed Finite Element Method ◽  
Immersed Finite Element ◽  
Elasticity Problems ◽  
Element Method

AbstractThe purpose of this paper is to develop a reduced Crouzeix–Raviart immersed finite element method (RCRIFEM) for two-dimensional elasticity problems with interface, which is based on the Kouhia–Stenberg finite element method (Kouhia et al. 1995) and Crouzeix–Raviart IFEM (CRIFEM) (Kwak et al. 2017). We use a {P_{1}}-conforming like element for one of the components of the displacement vector, and a {P_{1}}-nonconforming like element for the other component. The number of degrees of freedom of our scheme is reduced to two thirds of CRIFEM. Furthermore, we can choose penalty parameters independent of the Poisson ratio. One of the penalty parameters depends on Lamé’s second constant μ, and the other penalty parameter is independent of both μ and λ. We prove the optimal order error estimates in piecewise {H^{1}}-norm, which is independent of the Poisson ratio. Numerical experiments show optimal order of convergence both in {L^{2}} and piecewise {H^{1}}-norms for all problems including nearly incompressible cases.

Flexible Integration Points Coupled with Smoothed Strain in Elasticity Problems

2017 ◽  
Vol 09 (06) ◽  
pp. 1750079
Author(s):  
Eric Li ◽  
W. H. Liao
Keyword(s):  
Finite Element Method ◽  
Finite Element ◽  
Integration Rule ◽  
Quadrilateral Elements ◽  
Numerical Studies ◽  
Elasticity Problems ◽  
Flexible Integration ◽  
The Stability ◽  
Stiffening Effect ◽  
Element Method

In this paper, alpha finite element method ([Formula: see text]FEM) with modified integration rule ([Formula: see text]FEM-MIR) using quadrilateral elements is developed. The key feature of [Formula: see text]FEM-MIR is to combine the smoothed strain and compatible strain using flexible integration points. With simple adjustment of integration points in the stiffness, it is found that the softening or stiffening effect of [Formula: see text]FEM-MIR model can be altered. In addition, the exact, upper and lower bound solutions of strain energy in the [Formula: see text]FEM-MIR model with different integration points are examined for both overestimation and underestimation problems. Furthermore, the displacement solutions can be improved significantly compared with traditional integration points in the standard finite element method (FEM) and [Formula: see text]FEM models. In this work, the strategy to overcome the volumetric locking and hourglass issues are also analyzed using different integration points. In addition, it is found that the stability of discretized model is proportional to parameter [Formula: see text] ([Formula: see text]controls the locations of integration points of stiffness) in the [Formula: see text]FEM-MIR model. Extensive numerical studies have been conducted to confirm the properties of the proposed [Formula: see text]FEM-MIR, and an excellent performance has been observed in comparing traditional [Formula: see text]FEM and FEM.

scholarly journals A Consistent Immersed Finite Element Method for the Interface Elasticity Problems

2016 ◽  
Vol 2016 ◽  
pp. 1-9
Author(s):  
Sangwon Jin ◽  
Do Y. Kwak ◽  
Daehyeon Kyeong
Keyword(s):  
Finite Element Method ◽  
Finite Element ◽  
Rates Of Convergence ◽  
Optimal Error Estimates ◽  
Optimal Error ◽  
Immersed Finite Element Method ◽  
Elasticity Problems ◽  
Interface Elasticity ◽  
Nonconforming Finite Element Methods ◽  
Element Method

We propose a new scheme for elasticity problems having discontinuity in the coefficients. In the previous work (Kwak et al., 2014), the authors suggested a method for solving such problems by finite element method using nonfitted grids. The proposed method is based on theP1-nonconforming finite element methods with stabilizing terms. In this work, we modify the method by adding the consistency terms, so that the estimates of consistency terms are not necessary. We show optimal error estimates inH1and divergence norms under minimal assumptions. Various numerical experiments also show optimal rates of convergence.

Sign in / Sign up

Export Citation Format

Share Document