ESAIM: M2AN 51 (2017) 187–207 ESAIM: Mathematical Modelling and Numerical Analysis DOI: 10.1051/m2an/2016011 www.esaim-m2an.org A STABILIZED P 1 -NONCONFORMING IMMERSED FINITE ELEMENT METHOD FOR THE INTERFACE ELASTICITY PROBLEMS ∗ Do Y. Kwak 1 , Sangwon Jin 1 and Daehyeon Kyeong 1 Abstract. 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, NewYork (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 Lam` e parameters μ and λ and locking free as λ →∞. Mathematics Subject Classification. 65N30, 74S05, 74B05. Received May 18, 2015. Revised November 19, 2015. Accepted February 1st, 2016. 1. Introduction Linear elasticity equation plays an important role in solid mechanics. In particular, when an elastic body is occupied by heterogeneous materials having distinct Lam` e parameters μ and λ, the governing equation holds on each disjoint domain and certain jump conditions must be satisfied along the interface of two materials [20]. This kind of problems involving composite materials is getting more and more attentions from both engineers and mathematicians in recent years, but efficient numerical schemes are not fully developed yet. To solve such equations numerically, one usually uses finite element methods with meshes aligned with the interface between two materials. However, such methods involve unstructured grids resulting in algebraic systems which involve more unknowns and irregular data structure. Keywords and phrases. Immersed finite element method, Crouzeix–Raviart finite element, elasticity problems, heterogeneous materials, stability terms, Laplace–Young condition. ∗ NRF Grant No. 2014R1A2A1A11053889. 1 Department of Mathematical Sciences, Korea Advanced Institute of Science and Technology, Daejeon, Korea. [email protected]; [email protected]; [email protected] Article published by EDP Sciences c EDP Sciences, SMAI 2016
A STABILIZED P1-NONCONFORMING IMMERSED FINITE ELEMENT METHOD FOR THE INTERFACE ELASTICITY PROBLEMS
Jun 12, 2023
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