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.

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.

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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