Error estimates for a partially penalized immersed finite element method for elasticity interface problems

2020 ◽  
Vol 54 (1) ◽  
pp. 1-24 ◽  
Author(s):  
Ruchi Guo ◽  
Tao Lin ◽  
Yanping Lin
Keyword(s):  
Finite Element ◽  
Bilinear Form ◽  
Energy Norm ◽  
Optimal Error ◽  
Interface Elements ◽  
Immersed Finite Element Method ◽  
Immersed Finite Element ◽  
Planar Elasticity ◽  
Elasticity Problems ◽  
Optimal Error Bound

This article is about the error analysis for a partially penalized immersed finite element (PPIFE) method designed to solve linear planar-elasticity problems whose Lamé parameters are piecewise constants with an interface-independent mesh. The bilinear form in this method contains penalties to handle the discontinuity in the global immersed finite element (IFE) functions across interface edges. We establish a stress trace inequality for IFE functions on interface elements, we employ a patch idea to derive an optimal error bound for the stress of the IFE interpolation on interface edges, and we design a suitable energy norm by which the bilinear form in this PPIFE method is coercive. These key ingredients enable us to prove that this PPIFE method converges optimally in both an energy norm and the usual L2 norm under the standard piecewise H2-regularity assumption for the exact solution. Features of the proposed PPIFE method are demonstrated with numerical examples.

scholarly journals An Immersed Finite Element Method for Planar Elasticity Interface Problem

2020 ◽  
pp. 1-16
Author(s):  
Angran Liu
Keyword(s):  
Finite Element ◽  
A Priori ◽  
Interface Problem ◽  
Interface Elements ◽  
Immersed Finite Element Method ◽  
A Priori Error Estimate ◽  
Immersed Finite Element ◽  
Planar Elasticity ◽  
Discrete Formulation ◽  
Theoretical Results

This paper presents the P1/CR immersed finite element (IFE) method to solve planar elasticityinterface problem. By adding some stabilisation terms on the edges of interface elements, thestability of the discrete formulation and a priori error estimate in an energy norm are presented.Finally, numerical examples are given to confirm our theoretical results.

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.

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.

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.

Immersed Methods for High Reynolds Number Fluid-Structure Interactions

2017 ◽  
Vol 14 (06) ◽  
pp. 1750068 ◽  
Author(s):  
Lucy T. Zhang
Keyword(s):  
Finite Element Method ◽  
Reynolds Number ◽  
Finite Element ◽  
High Reynolds Number ◽  
Fluid Structure Interactions ◽  
Immersed Finite Element Method ◽  
Immersed Finite Element ◽  
High Reynolds ◽  
Immersed Methods ◽  
Reynolds Number Flows

Immersed methods are considered as a class of nonboundary-fitted meshing technique for simulating fluid–structure interactions. However, the conventional approach of coupling the fluid and solid domains, as in the immersed boundary method and the immersed finite element method, often cannot handle high Reynolds number flows interacting with moving and deformable solids. As the solid dynamics is imposed by the fluid dynamics, it often leads to unrealistically large deformation of the solid in cases of high Reynolds number flows. The first attempt in resolving this issue was proposed in the modified immersed finite element method (mIFEM), however, some terms were determined heuristically. In this paper, we provide a full and rigorous derivation for the mIFEM with corrections to the previously proposed terms, which further extends the accuracy of the algorithm. In the “swapped” coupling logic, we solve for the dynamics of the solid, and then numerically impose it to the background fluid, which allows the solid to control its own dynamics and governing laws instead of following that of the fluid. A few examples including a biomedical engineering application are shown to demonstrate the capability in handling large Reynolds number flows using the derived mIFEM.

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