Volume 13, Issue 2
A Stabilized Low Order Finite Element Method for Three Dimensional Elasticity Problems

Numer. Math. Theor. Meth. Appl., 13 (2020), pp. 281-295.

Published online: 2020-03

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• Abstract

We introduce a low order finite element method for three dimensional elasticity problems. We extend Kouhia-Stenberg element [12] by using two nonconforming components and one conforming component, adding stabilizing terms to the associated bilinear form to ensure the discrete Korn's inequality. Using the second Strang's lemma, we show that our scheme has optimal convergence rates in $L^2$ and piecewise $H^1$-norms even when Poisson ratio $\nu$ approaches $1/2$. Even though some efforts have been made to design a low order method for three dimensional problems in [11,16], their method uses some higher degree basis functions. Our scheme is the first true low order method. We provide three numerical examples which support our analysis. We compute two examples having analytic solutions. We observe the optimal $L^2$ and $H^1$ errors for many different choices of Poisson ratios including the nearly incompressible cases. In the last example, we simulate the driven cavity problem. Our scheme shows non-locking phenomena for the driven cavity problems also.

• Keywords

Elasticity equation, low order finite element, Kouhia-Stenberg element, locking free, Korn's inequality.

65N12, 65N30

gwanghyun@kunsan.ac.kr (Gwanghyun Jo)

kdy@kaist.ac.kr (Do Y. Kwak)

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@Article{NMTMA-13-281, author = {Jo , Gwanghyun and Kwak , Do Y. }, title = {A Stabilized Low Order Finite Element Method for Three Dimensional Elasticity Problems}, journal = {Numerical Mathematics: Theory, Methods and Applications}, year = {2020}, volume = {13}, number = {2}, pages = {281--295}, abstract = {

We introduce a low order finite element method for three dimensional elasticity problems. We extend Kouhia-Stenberg element [12] by using two nonconforming components and one conforming component, adding stabilizing terms to the associated bilinear form to ensure the discrete Korn's inequality. Using the second Strang's lemma, we show that our scheme has optimal convergence rates in $L^2$ and piecewise $H^1$-norms even when Poisson ratio $\nu$ approaches $1/2$. Even though some efforts have been made to design a low order method for three dimensional problems in [11,16], their method uses some higher degree basis functions. Our scheme is the first true low order method. We provide three numerical examples which support our analysis. We compute two examples having analytic solutions. We observe the optimal $L^2$ and $H^1$ errors for many different choices of Poisson ratios including the nearly incompressible cases. In the last example, we simulate the driven cavity problem. Our scheme shows non-locking phenomena for the driven cavity problems also.

}, issn = {2079-7338}, doi = {https://doi.org/10.4208/nmtma.OA-2019-0034}, url = {http://global-sci.org/intro/article_detail/nmtma/15443.html} }
TY - JOUR T1 - A Stabilized Low Order Finite Element Method for Three Dimensional Elasticity Problems AU - Jo , Gwanghyun AU - Kwak , Do Y. JO - Numerical Mathematics: Theory, Methods and Applications VL - 2 SP - 281 EP - 295 PY - 2020 DA - 2020/03 SN - 13 DO - http://doi.org/10.4208/nmtma.OA-2019-0034 UR - https://global-sci.org/intro/article_detail/nmtma/15443.html KW - Elasticity equation, low order finite element, Kouhia-Stenberg element, locking free, Korn's inequality. AB -

We introduce a low order finite element method for three dimensional elasticity problems. We extend Kouhia-Stenberg element [12] by using two nonconforming components and one conforming component, adding stabilizing terms to the associated bilinear form to ensure the discrete Korn's inequality. Using the second Strang's lemma, we show that our scheme has optimal convergence rates in $L^2$ and piecewise $H^1$-norms even when Poisson ratio $\nu$ approaches $1/2$. Even though some efforts have been made to design a low order method for three dimensional problems in [11,16], their method uses some higher degree basis functions. Our scheme is the first true low order method. We provide three numerical examples which support our analysis. We compute two examples having analytic solutions. We observe the optimal $L^2$ and $H^1$ errors for many different choices of Poisson ratios including the nearly incompressible cases. In the last example, we simulate the driven cavity problem. Our scheme shows non-locking phenomena for the driven cavity problems also.

Gwanghyun Jo & Do Y. Kwak. (2020). A Stabilized Low Order Finite Element Method for Three Dimensional Elasticity Problems. Numerical Mathematics: Theory, Methods and Applications. 13 (2). 281-295. doi:10.4208/nmtma.OA-2019-0034
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