A stabilized P1-nonconforming immersed finite element method for the interface elasticity problems
Kwak, Do Y.1 ; Jin, Sangwon1 ; Kyeong, Daehyeon1
1 Department of Mathematical Sciences, Korea Advanced Institute of Science and Technology, Daejeon, Korea.
ESAIM: Mathematical Modelling and Numerical Analysis , Tome 51 (2017) no. 1, pp. 187-207.

We develop a new finite element method for solving planar elasticity problems involving heterogeneous materials with a mesh not necessarily aligning with the interface of the materials. This method is based on the ‘broken’ Crouzeix–Raviart P1-nonconforming finite element method for elliptic interface problems [D.Y. Kwak, K.T. Wee and K.S. Chang, SIAM J. Numer. Anal. 48 (2010) 2117–2134]. To ensure the coercivity of the bilinear form arising from using the nonconforming finite elements, we add stabilizing terms as in the discontinuous Galerkin (DG) method [D.N. Arnold, SIAM J. Numer. Anal. 19 (1982) 742–760; D.N. Arnold and F. Brezzi, in Discontinuous Galerkin Methods. Theory, Computation and Applications, edited by B. Cockburn, G.E. Karniadakis, and C.-W. Shu. Vol. 11 of Lecture Notes in Comput. Sci. Engrg. Springer-Verlag, New York (2000) 89–101; M.F. Wheeler, SIAM J. Numer. Anal. 15 (1978) 152–161.]. The novelty of our method is that we use meshes independent of the interface, so that the interface may cut through the elements. Instead, we modify the basis functions so that they satisfy the Laplace–Young condition along the interface of each element. We prove optimal H1 and divergence norm error estimates. Numerical experiments are carried out to demonstrate that our method is optimal for various Lamè parameters μ and λ and locking free as λ→∞.

DOI : 10.1051/m2an/2016011
Classification : 65N30, 74S05, 74B05
Mots-clés : Immersed finite element method, Crouzeix–Raviart finite element, elasticity problems, heterogeneous materials, stability terms, Laplace–Young condition
Kwak, Do Y. 1 ; Jin, Sangwon 1 ; Kyeong, Daehyeon 1

1 Department of Mathematical Sciences, Korea Advanced Institute of Science and Technology, Daejeon, Korea. 
@article{M2AN_2017__51_1_187_0,
     author = {Kwak, Do Y. and Jin, Sangwon and Kyeong, Daehyeon},
     title = {A stabilized $P_{1}$-nonconforming immersed finite element method for the interface elasticity problems},
     journal = {ESAIM: Mathematical Modelling and Numerical Analysis },
     pages = {187--207},
     publisher = {EDP-Sciences},
     volume = {51},
     number = {1},
     year = {2017},
     doi = {10.1051/m2an/2016011},
     zbl = {1381.74199},
     language = {en},
     url = {https://www.numdam.org/articles/10.1051/m2an/2016011/}
}
TY  - JOUR
AU  - Kwak, Do Y.
AU  - Jin, Sangwon
AU  - Kyeong, Daehyeon
TI  - A stabilized $P_{1}$-nonconforming immersed finite element method for the interface elasticity problems
JO  - ESAIM: Mathematical Modelling and Numerical Analysis 
PY  - 2017
SP  - 187
EP  - 207
VL  - 51
IS  - 1
PB  - EDP-Sciences
UR  - https://www.numdam.org/articles/10.1051/m2an/2016011/
DO  - 10.1051/m2an/2016011
LA  - en
ID  - M2AN_2017__51_1_187_0
ER  - 
%0 Journal Article
%A Kwak, Do Y.
%A Jin, Sangwon
%A Kyeong, Daehyeon
%T A stabilized $P_{1}$-nonconforming immersed finite element method for the interface elasticity problems
%J ESAIM: Mathematical Modelling and Numerical Analysis 
%D 2017
%P 187-207
%V 51
%N 1
%I EDP-Sciences
%U https://www.numdam.org/articles/10.1051/m2an/2016011/
%R 10.1051/m2an/2016011
%G en
%F M2AN_2017__51_1_187_0
Kwak, Do Y.; Jin, Sangwon; Kyeong, Daehyeon. A stabilized $P_{1}$-nonconforming immersed finite element method for the interface elasticity problems. ESAIM: Mathematical Modelling and Numerical Analysis , Tome 51 (2017) no. 1, pp. 187-207. doi : 10.1051/m2an/2016011. https://www.numdam.org/articles/10.1051/m2an/2016011/

D.N. Arnold, An interior penalty finite element method with discontinuous elements. SIAM J. Numer. Anal. 19 (1982) 742–760. | DOI | MR | Zbl

D.N. Arnold and R. Winther, Mixed finite elements for elasticity. Numer. Math. 92 (2002) 401–419. | DOI | MR | Zbl

D.N. Arnold, F. Brezzi, B. Cockburn and D. Marini, Discontinuous Galerkin methods for elliptic problems, in Discontinuous Galerkin Methods. Theory. In vol. 11 of Computation and Applications, edited by B. Cockburn, G.E. Karniadakis and C.-W. Shu. Lecture Notes Comput. Sci. Engrg. Springer-Verlag, New York (2000) 89–101. | Zbl

I. Babuška and M. Suri, Locking effect in the finite element approximation of elasticity problem. Numer. Math. 62 (1992) 439–463. | DOI | Zbl

I. Babuška and M. Suri, On locking and robustness in the finie element method. SIAM J. Numer. Anal. 29 (1992) 1261–1293. | DOI | Zbl

R. Becker, E. Burman and P. Hansbo, A Nitsche extended finite element method for incompressible elasticity with discontinuous modulus of elasticity. Comput. Methods Appl. Mech. Engrg. 198 (2009) 3352–3360. | DOI | Zbl

T. Belytschko and T. Black, Elastic crack growth in finite elements with minimal remeshing. Int. J. Numer. Meth. Engrg. 45 (1999) 601–620. | DOI | Zbl

T. Belytschko, C. Parimi, N. Moës, N. Sukumar and S. Usui, Structured extended finite element methods for solids defined by implicit surfaces. Int. J. Numer. Meth. Engrg. 56 (2003) 609–635. | DOI | Zbl

D. Braess, Finite elements: Theory, fast solvers, and applications in solid mechanics, 2nd edition. Cambridge University Press, Cambridge (2001). | Zbl

S.C. Brenner, Korn’s inequalities for piecewise H1 vector fields. Math. Comp. 72 (2003) 1067–1087. | DOI | Zbl

S.C. Brenner and L.Y. Sung, Linear finite element methods for planar linear elasticity. Math. Comp. 59 (1992) 321–338. | DOI | Zbl

F. Brezzi and M. Fortin, Mixed and hybrid finite element methods. Springer-Verlag, New-York (1991). | Zbl

K.S. Chang and D.Y. Kwak, Discontinuous Bubble scheme for elliptic problems with jumps in the solution. Comput. Method Appl. Mech. Engrg. 200 (2011) 494–508. | DOI | Zbl

S.H. Chou, D.Y. Kwak and K.T. Wee, Optimal convergence analysis of an immersed interface finite element method. Adv. Comput. Math. 33 (2010) 149–168. | DOI | Zbl

P.G. Ciarlet, The finite element method for elliptic problems. North Holland (1978). | Zbl

P.G. Ciarlet, Mathematical elasticity. Vol I. North Holland (1988). | Zbl

M. Crouzeix and P.A. Raviart, Conforming and nonconforming finite element methods for solving the stationary Stokes equations. RAIRO Anal. Numér. 7 (1973) 33–75. | Numdam | Zbl

R.S. Falk, Nonconforming Finite Element Methods for the Equations of Linear Elasticity. Math. Comput. 57 (1991) 529–550. | DOI | Zbl

A. Hansbo and P. Hansbo, An unfitted finite element method, based on Nitsche’s method, for elliptic interface problems. Comput. Methods Appl. Mech. Engrg. 191 (2002) 5537–5552. | DOI | Zbl

A. Hansbo and P. Hansbo, A finite element method for the simulation of strong and weak discontinuities in solid mechanics. Comput. Methods Appl. Mech. Engrg. 193 (2004) 3523–3540. | DOI | Zbl

P. Hansbo and M.G. Larson, Discontinuous Galerkin and the Crouzeix–Raviart element: Applications to elasticity. ESAIM: M2AN 37 (2003) 63–72. | DOI | Numdam | Zbl

P. Krysl and T. Belytschko, An effcient linear-precision partition of unity basis for unstructured meshless methods. Commun. Numer. Meth. Engng. 16 (2000) 239–255. | DOI | Zbl

D.Y. Kwak, K.T. Wee and K.S. Chang, An analysis of a broken P1 -nonconforming finite element method for interface problems. SIAM J. Numer. Anal. 48 (2010) 2117–2134. | DOI | Zbl

M. Lai, Z. Li and X. Lin, Fast solvers for 3D Poisson equations involving interfaces in a finite or the infinite domain. J. Comput. Appl. Math. 191 (2006) 106–125. | DOI | Zbl

G. Legrain, N. Moës and E. Verron, Stress analysis around crack tips in finite strain problems using the eXtended finite element method. Int. J. Numer. Meth. Eng. 63 (2005) 290–314. | DOI | Zbl

D. Leguillon and E. Sanchez-Palencia, Computation of Singular Solutions in Elliptic Problems and Elasticity. Wiley (1987). | Zbl

R.J. Leveque and Z. Li, The immersed interface method for elliptic equations with discontinuous coefficients and singular sources. SIAM J. Numer. Anal. 31 (1994) 1019–1044. | DOI | Zbl

R.J. Leveque and Z. Li, Immersed interface method for Stokes flow with elastic boundaries or surface tension. SIAM J. Sci. Comput. 18 (1997) 709–735. | DOI | Zbl

T. Lin and X. Zhang, Linear and bilinear immersed finite elements for planar elasticity interface problems. J. Comput. Appl. Math. 236 (2012) 4681–4699. | DOI | Zbl

Z. Li, T. Lin and X. Wu, New Cartesian grid methods for interface problems using the finite element formulation. Numer. Math. 96 (2003) 61–98. | DOI | Zbl

Z. Li, T. Lin, Y. Lin and R.C. Rogers, An immersed finite element space and its approximation capability. Numer. Methods Partial Differ. Eq. 20 (2004) 338–367. | DOI | Zbl

T. Lin, D. Sheen and X. Zhang, A locking-free immersed finite element method for planar elasticity interface problems. J. Comput. Phys. 247 (2013) 228–247. | DOI | Zbl

N. Moës, J. Dolbow and T. Belytschko, A finite element method for crack growth without remeshing. Int. J. Numer. Methods Eng. 46 (1999) 131–156. | DOI | Zbl

J. Nitsche, Über ein Variationsprinzip zur Lösung von Dirichlet-Problemen bei Verwendung von Teilraumen die keinen Randbedingungen unterworfen sind. Abh. Math. Sem. Univ. Hamburg 36 (1971) 9–15. | DOI | Zbl

M. Oevermann, C. Scharfenberg and R. Klein, A sharp interface finite volume method for elliptic equations on Cartesian grids. J. Comput. Phys. 228 (2009) 5184–5206. | DOI | Zbl

M.F. Wheeler, An elliptic collocation-finite element method with interior penalties. SIAM J. Numer. Anal. 15 (1978) 152–161. | DOI | Zbl

  • Han, Yajie; Xia, Guangqing; Kang, Huifeng; Lu, Chang; Chen, Chong; Saetchnikov, Vladimir Three-Dimensional Direct-Implicit Particle-in-Cell Model Using Trilinear Anisotropic Immersed-Finite-Element for Plasma Propulsion, AIAA Journal, Volume 63 (2025) no. 1, p. 323 | DOI:10.2514/1.j064305
  • Jo, Gwanghyun; Kwak, Do Young A New Immersed Finite Element Method for Two-Phase Stokes Problems Having Discontinuous Pressure, Computational Methods in Applied Mathematics, Volume 24 (2024) no. 1, p. 49 | DOI:10.1515/cmam-2022-0122
  • Yang, Fanyi The least squares finite element method for elasticity interface problem on unfitted mesh, ESAIM: Mathematical Modelling and Numerical Analysis, Volume 58 (2024) no. 2, p. 695 | DOI:10.1051/m2an/2024015
  • Wu, Siyu; Bai, Jinwei; He, Xiaoming; Zhao, Ren; Cao, Yong An immersed selective discontinuous Galerkin method in particle-in-cell simulation with adaptive Cartesian mesh and polynomial preserving recovery, Journal of Computational Physics, Volume 498 (2024), p. 112703 | DOI:10.1016/j.jcp.2023.112703
  • Han, Yajie; Xia, Guangqing; Lu, Chang; He, Xiaoming Trilinear Immersed-Finite-Element Method for Three-Dimensional Anisotropic Interface Problems in Plasma Thrusters, AIAA Journal, Volume 61 (2023) no. 10, p. 4267 | DOI:10.2514/1.j062614
  • Jo, Gwanghyun; Kwak, Do Y. Immersed finite element methods for convection diffusion equations, AIMS Mathematics, Volume 8 (2023) no. 4, p. 8034 | DOI:10.3934/math.2023407
  • Park, Hyeokjoo; Kwak, Do Y. An immersed weak Galerkin method for elliptic interface problems on polygonal meshes, Computers Mathematics with Applications, Volume 147 (2023), p. 185 | DOI:10.1016/j.camwa.2023.07.025
  • Tang, Zhongliang; Zheng, Yu; Wang, Liqun; Wang, Quanxiang A new Petrov–Galerkin immersed finite element method for elliptic interface problems with non-homogeneous jump conditions, Journal of Engineering Mathematics, Volume 141 (2023) no. 1 | DOI:10.1007/s10665-023-10286-3
  • Choi, Yoonjeong; Jo, Gwanghyun; Kwak, Do Y.; Lee, Young Ju Locally conservative discontinuous bubble scheme for Darcy flow and its application to Hele-Shaw equation based on structured grids, Numerical Algorithms, Volume 92 (2023) no. 2, p. 1127 | DOI:10.1007/s11075-022-01333-8
  • Jamei, Mehdi; Mosharaf-Dehkordi, Mehdi; Ghafouri, Hamid Reza A Sequentially- Hybridized Locally Conservative Non-conforming Finite Element Scheme for Two-phase Flow Simulation through Heterogeneous Porous Media, Advances in Water Resources, Volume 162 (2022), p. 104155 | DOI:10.1016/j.advwatres.2022.104155
  • Wang, Quanxiang; Xie, Jianqiang; Zhang, Zhiyue Nonconforming immersed finite element method for solving elliptic optimal control problems with interfaces, Applicable Analysis, Volume 101 (2022) no. 6, p. 2197 | DOI:10.1080/00036811.2020.1802431
  • Kwak, Do Y.; Park, Hyeokjoo Lowest-order virtual element methods for linear elasticity problems, Computer Methods in Applied Mechanics and Engineering, Volume 390 (2022), p. 114448 | DOI:10.1016/j.cma.2021.114448
  • Zhang, Xiaodi High order interface-penalty finite element methods for elasticity interface problems in 3D, Computers Mathematics with Applications, Volume 114 (2022), p. 161 | DOI:10.1016/j.camwa.2022.03.044
  • Kaudur, Srivatsa Bhat; Patil, Mayuresh J. Shape optimization with immersed interface finite element method, International Journal for Numerical Methods in Engineering, Volume 123 (2022) no. 23, p. 5907 | DOI:10.1002/nme.7093
  • Jeon, Youngmok An immersed hybrid difference method for the elliptic interface equation, Japan Journal of Industrial and Applied Mathematics, Volume 39 (2022) no. 2, p. 669 | DOI:10.1007/s13160-022-00503-4
  • Kwon, In; Jo, Gwanghyun; Shin, Kwang-Seong A Deep Neural Network Based on ResNet for Predicting Solutions of Poisson–Boltzmann Equation, Electronics, Volume 10 (2021) no. 21, p. 2627 | DOI:10.3390/electronics10212627
  • Gholampour, Faranak; Hesameddini, Esmail; Taleei, Ameneh A stable RBF partition of unity local method for elliptic interface problems in two dimensions, Engineering Analysis with Boundary Elements, Volume 123 (2021), p. 220 | DOI:10.1016/j.enganabound.2020.10.016
  • Kwon, In; Kwak, Do Y.; Jo, Gwanghyun Discontinuous bubble immersed finite element method for Poisson-Boltzmann-Nernst-Planck model, Journal of Computational Physics, Volume 438 (2021), p. 110370 | DOI:10.1016/j.jcp.2021.110370
  • Huang, Peiqi; Li, Zhilin Partially penalized IFE methods and convergence analysis for elasticity interface problems, Journal of Computational and Applied Mathematics, Volume 382 (2021), p. 113059 | DOI:10.1016/j.cam.2020.113059
  • Jo, Gwanghyun; Kwak, Do Y.; Lee, Young-Ju Locally Conservative Immersed Finite Element Method for Elliptic Interface Problems, Journal of Scientific Computing, Volume 87 (2021) no. 2 | DOI:10.1007/s10915-021-01476-1
  • Han, Daoru; He, Xiaoming; Lund, David; Zhang, Xu PIFE-PIC: Parallel Immersed Finite Element Particle-in-Cell for 3-D Kinetic Simulations of Plasma-Material Interactions, SIAM Journal on Scientific Computing, Volume 43 (2021) no. 3, p. C235 | DOI:10.1137/20m137344x
  • Jo, Gwanghyun; Kwak, Do Young A Reduced Crouzeix–Raviart Immersed Finite Element Method for Elasticity Problems with Interfaces, Computational Methods in Applied Mathematics, Volume 20 (2020) no. 3, p. 501 | DOI:10.1515/cmam-2019-0046
  • Lu, Chang; Wan, Jie; Cao, Yong; He, Xiaoming A fully decoupled iterative method with three-dimensional anisotropic immersed finite elements for Kaufman-type discharge problems, Computer Methods in Applied Mechanics and Engineering, Volume 372 (2020), p. 113345 | DOI:10.1016/j.cma.2020.113345
  • Lu, Chang; Yang, Zhi; Bai, Jinwei; Cao, Yong; He, Xiaoming Three‐dimensional immersed finite‐element method for anisotropic magnetostatic/electrostatic interface problems with nonhomogeneous flux jump, International Journal for Numerical Methods in Engineering, Volume 121 (2020) no. 10, p. 2107 | DOI:10.1002/nme.6301
  • WANG, Q.; ZHANG, Z. A MODIFIED IMMERSED FINITE VOLUME ELEMENT METHOD FOR ELLIPTIC INTERFACE PROBLEMS, The ANZIAM Journal, Volume 62 (2020) no. 1, p. 42 | DOI:10.1017/s1446181120000073
  • Wang, Quanxiang; Zhang, Zhiyue A stabilized immersed finite volume element method for elliptic interface problems, Applied Numerical Mathematics, Volume 143 (2019), p. 75 | DOI:10.1016/j.apnum.2019.03.010
  • Kyeong, Daehyeon; Kwak, Do Young An Immersed Finite Element Method for the Elasticity Problems with Displacement Jump, Advances in Applied Mathematics and Mechanics, Volume 9 (2017) no. 2, p. 407 | DOI:10.4208/aamm.2016.m1427
  • Jin, Sangwon; Kwak, Do Y.; Kyeong, Daehyeon A Consistent Immersed Finite Element Method for the Interface Elasticity Problems, Advances in Mathematical Physics, Volume 2016 (2016), p. 1 | DOI:10.1155/2016/3292487

Cité par 28 documents. Sources : Crossref