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.

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.

A Stabilized Finite Element Method for Non-Stationary Conduction-Convection Problems

2011 ◽  
Vol 3 (2) ◽  
pp. 239-258 ◽  
Author(s):  
Ke Zhao ◽  
Yinnian He ◽  
Tong Zhang
Keyword(s):  
Finite Element Method ◽  
Finite Element ◽  
Optimal Error Estimates ◽  
Two Dimensional ◽  
Element Level ◽  
Optimal Error ◽  
Stabilized Finite Element ◽  
Stabilized Finite Element Method ◽  
The Stability ◽  
Element Method

AbstractThis paper is concerned with a stabilized finite element method based on two local Gauss integrations for the two-dimensional non-stationary conduction-convection equations by using the lowest equal-order pairs of finite elements. This method only offsets the discrete pressure space by the residual of the simple and symmetry term at element level in order to circumvent the inf-sup condition. The stability of the discrete scheme is derived under some regularity assumptions. Optimal error estimates are obtained by applying the standard Galerkin techniques. Finally, the numerical illustrations agree completely with the theoretical expectations.

An Immersed Finite Element Method for the Elasticity Problems with Displacement Jump

2017 ◽  
Vol 9 (2) ◽  
pp. 407-428 ◽  
Author(s):  
Daehyeon Kyeong ◽  
Do Young Kwak
Keyword(s):  
Finite Element Method ◽  
Finite Element ◽  
Normal Stress ◽  
Basis Functions ◽  
Displacement Discontinuity ◽  
Immersed Finite Element Method ◽  
Immersed Finite Element ◽  
Uniform Grids ◽  
Elasticity Problems ◽  
Element Method

AbstractIn this paper, we propose a finite element method for the elasticity problems which have displacement discontinuity along the material interface using uniform grids. We modify the immersed finite element method introduced recently for the computation of interface problems having homogeneous jumps [20, 22]. Since the interface is allowed to cut through the element, we modify the standard Crouzeix-Raviart basis functions so that along the interface, the normal stress is continuous and the jump of the displacement vector is proportional to the normal stress. We construct the broken piecewise linear basis functions which are uniquely determined by these conditions. The unknowns are only associated with the edges of element, except the intersection points. Thus our scheme has fewer degrees of freedom than most of the XFEM type of methods in the existing literature [1,8,13]. Finally, we present numerical results which show optimal orders of convergence rates.

Non-conforming Computational Methods for Mixed Elasticity Problems

2003 ◽  
Vol 3 (1) ◽  
pp. 23-34
Author(s):  
Faker Ben Belgacem ◽  
Lawrence K. Chilton ◽  
Padmanabhan Seshaiyer
Keyword(s):  
Finite Element Method ◽  
Finite Element ◽  
Mixed Finite Elements ◽  
Rates Of Convergence ◽  
Modeling Methodology ◽  
Conforming Finite Element ◽  
Elasticity Problems ◽  
Conforming Finite Element Method ◽  
Incompressible Elasticity ◽  
Element Method

AbstractIn this paper, we present a non-conforming hp computational modeling methodology for solving elasticity problems. We consider the incompressible elasticity model formulated as a mixed displacement-pressure problem on a global domain which is partitioned into several local subdomains. The approximation within each local subdomain is designed using div-stable hp-mixed finite elements. The displacement is computed in a mortared space while the pressure is not subjected to any constraints across the interfaces. Our computational results demonstrate that the non-conforming finite element method presented for the elasticity problem satisfies similar rates of convergence as the conforming finite element method, in the presence of various h-version and p-version discretizations.

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