A stabilized $P_1$-nonconforming immersed finite element method for the interface elasticity problems

Kwak, Do Y.; Jin, Sangwon; Kyeong, Daehyeon

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 $P_1$-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 $H^1$ and divergence norm error estimates. Numerical experiments are carried out to demonstrate that our method is optimal for various Lamè parameters $\mu$ and $\lambda$ and locking free as $\lambda \to \infty$.

Classification : 65N30, 74S05, 74B05
Keywords: Immersed finite element method, Crouzeix–Raviart finite element, elasticity problems, heterogeneous materials, stability terms, Laplace–Young condition

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

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

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. | Zbl | DOI

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

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. | Zbl | DOI

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

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. | Zbl | DOI

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 $H^1$ vector fields. Math. Comp. 72 (2003) 1067–1087. | Zbl | DOI

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

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. | Zbl | DOI

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. | Zbl | DOI

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. | Zbl | Numdam

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

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. | Zbl | DOI

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. | Zbl | DOI

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

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. | Zbl | DOI

D.Y. Kwak, K.T. Wee and K.S. Chang, An analysis of a broken $P_1$ -nonconforming finite element method for interface problems. SIAM J. Numer. Anal. 48 (2010) 2117–2134. | Zbl | DOI

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. | Zbl | DOI

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. | Zbl | DOI

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. | Zbl | DOI

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. | Zbl | DOI

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

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. | Zbl | DOI

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. | Zbl | DOI

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. | Zbl | DOI

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. | Zbl | DOI

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. | Zbl | DOI

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. | Zbl | DOI

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