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Volume 9, Issue 2
An Immersed Finite Element Method for the Elasticity Problems with Displacement Jump

Daehyeon Kyeong & Do Young Kwak

Adv. Appl. Math. Mech., 9 (2017), pp. 407-428.

Published online: 2018-05

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  • Abstract

In 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.

  • Keywords

Elasticity problems, finite element method, Crouzeix-Raviart element, displacement discontinuity.

  • AMS Subject Headings

65N30, 74S05, 74B05

  • Copyright

COPYRIGHT: © Global Science Press

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@Article{AAMM-9-407, author = {Daehyeon and Kyeong and and 19526 and and Daehyeon Kyeong and Do Young and Kwak and and 19527 and and Do Young Kwak}, title = {An Immersed Finite Element Method for the Elasticity Problems with Displacement Jump}, journal = {Advances in Applied Mathematics and Mechanics}, year = {2018}, volume = {9}, number = {2}, pages = {407--428}, abstract = {

In 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.

}, issn = {2075-1354}, doi = {https://doi.org/10.4208/aamm.2016.m1427}, url = {http://global-sci.org/intro/article_detail/aamm/12156.html} }
TY - JOUR T1 - An Immersed Finite Element Method for the Elasticity Problems with Displacement Jump AU - Kyeong , Daehyeon AU - Kwak , Do Young JO - Advances in Applied Mathematics and Mechanics VL - 2 SP - 407 EP - 428 PY - 2018 DA - 2018/05 SN - 9 DO - http://doi.org/10.4208/aamm.2016.m1427 UR - https://global-sci.org/intro/article_detail/aamm/12156.html KW - Elasticity problems, finite element method, Crouzeix-Raviart element, displacement discontinuity. AB -

In 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.

Daehyeon Kyeong & Do Young Kwak. (2020). An Immersed Finite Element Method for the Elasticity Problems with Displacement Jump. Advances in Applied Mathematics and Mechanics. 9 (2). 407-428. doi:10.4208/aamm.2016.m1427
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